1787 lines
49 KiB
Plaintext
1787 lines
49 KiB
Plaintext
# Derivatives
|
||
|
||
|
||
{{< include ../_common_code.qmd >}}
|
||
|
||
This section uses these add-on packages:
|
||
|
||
|
||
```{julia}
|
||
using CalculusWithJulia
|
||
using Plots
|
||
plotly()
|
||
using SymPy
|
||
```
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
#| results: "hidden"
|
||
using DataFrames
|
||
nothing
|
||
```
|
||
|
||
---
|
||
|
||
::: {#fig-picture-of-galileo-ramp}
|
||
{width=60%}
|
||
|
||
Device to measure units of distance by units of time
|
||
:::
|
||
|
||
|
||
Before defining the derivative of a function, let's begin with two motivating examples.
|
||
|
||
|
||
##### Example: Driving
|
||
|
||
|
||
Imagine motoring along down highway $61$ leaving Minnesota on the way to New Orleans; though lost in listening to music, still mindful of the speedometer and odometer, both prominently placed on the dashboard of the car.
|
||
|
||
|
||
The speedometer reads $60$ miles per hour, what is the odometer doing? Besides recording total distance traveled, it is incrementing dutifully every hour by $60$ miles. Why? Well, the well-known formula relating distance, time and rate of travel is
|
||
|
||
|
||
$$
|
||
\text{distance} = \text{ rate } \times \text{ time.}
|
||
$$
|
||
|
||
If the rate is a constant $60$ miles/hour, then in one hour the distance traveled is $60$ miles.
|
||
|
||
|
||
Of course, the odometer isn't just incrementing once per hour, it is incrementing once every $1/10$th of a mile. How much time does that take? Well, we would need to solve $1/10=60 \cdot t$ which means $t=1/600$ hours, better known as once every $6$ seconds.
|
||
|
||
|
||
Using some mathematical notation, would give $x(t) = v\cdot t$, where $x$ is position at time $t$, $v$ is the *constant* velocity and $t$ the time traveled in hours. The simple graph of @fig-plot-position-time-over-0-3 shows the first three hours of travel.
|
||
|
||
|
||
::: {#fig-plot-position-time-over-0-3}
|
||
```{julia}
|
||
#| hold: true
|
||
position(t) = 60 * t
|
||
plot(position, 0, 3)
|
||
```
|
||
Plot of position versus time for a constant speed
|
||
:::
|
||
|
||
Oh no, we hit traffic. In the next $30$ minutes we only traveled $15$ miles. We were so busy looking out for traffic, the speedometer was not checked. What would the average speed have been? Though in the $30$ minutes of stop-and-go traffic, the displayed speed may have varied, the *average speed* would simply be the change in distance over the change in time, or $\Delta x / \Delta t$. That is
|
||
|
||
|
||
```{julia}
|
||
15/(1/2)
|
||
```
|
||
|
||
Now suppose that after $6$ hours of travel the GPS in the car gives us a readout of distance traveled as a function of time. @fig-plot-complicated-position-versus-time-from-0-to-6 shows the graph.
|
||
|
||
::: {#fig-plot-complicated-position-versus-time-from-0-to-6}
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
function position(t)
|
||
t <= 3 ? 60*t :
|
||
t <= 3.5 ? position(3) + 30*(t-3) :
|
||
t <= 4 ? position(3.5) + 75 * (t-3.5) :
|
||
t <= 4.5 ? position(4) : position(4.5) + 60*(t-4.5)
|
||
end
|
||
plot(position, 0, 6)
|
||
```
|
||
|
||
Plot of position versus time for a certain non constant speed
|
||
:::
|
||
|
||
We can see with some effort that the slope is steady for the first three hours, is slightly less between $3$ and $3.5$ hours, then is a bit steeper for the next half hour. After that, it is flat for the about half an hour, then the slope continues on with same value as in the first $3$ hours. What does that say about our speed during our trip?
|
||
|
||
|
||
Based on the graph, what was the average speed over the first three hours? Well, we traveled $180$ miles, and took $3$ hours:
|
||
|
||
|
||
```{julia}
|
||
180/3
|
||
```
|
||
|
||
What about the next half hour? Squinting shows the amount traveled was $15$ miles ($195 - 180$) and it took $1/2$ an hour:
|
||
|
||
|
||
```{julia}
|
||
15/(1/2)
|
||
```
|
||
|
||
And the half hour after that? The average speed is found from the distance traveled, $37.5$ miles, divided by the time, $1/2$ hour:
|
||
|
||
|
||
```{julia}
|
||
37.5 / (1/2)
|
||
```
|
||
|
||
Okay, so there was some speeding involved.
|
||
|
||
|
||
The next half hour the car did not move. What was the average speed? Well the change in position was $0$, but the time was $1/2$ hour, so the average was $0$.
|
||
|
||
|
||
Perhaps a graph of the speed is a bit more clear. We can do this based on the above in @fig-plot-non-constant-speed-over-0-6.
|
||
|
||
::: {#fig-plot-non-constant-speed-over-0-6}
|
||
```{julia}
|
||
function speed(t)
|
||
0 < t <= 3 ? 60 :
|
||
t <= 3.5 ? 30 :
|
||
t <= 4 ? 75 :
|
||
t <= 4.5 ? 0 : 60
|
||
end
|
||
plot(speed, 0, 6)
|
||
```
|
||
|
||
Plot of non-constant speed over $[0,6]$
|
||
:::
|
||
|
||
|
||
The jumps, as discussed before, are artifacts of the graphing algorithm. What is interesting, is we could have derived the graph of `speed` from that of `x` by just finding the slopes of the line segments, and we could have derived the graph of `x` from that of `speed`, just using the simple formula relating distance, rate, and time.
|
||
|
||
|
||
:::{.callout-note}
|
||
## Note
|
||
We were pretty loose with some key terms. There is a distinction between "speed" and "velocity", this being the speed is the absolute value of velocity. Velocity incorporates a direction as well as a magnitude. Similarly, distance traveled and change in position are not the same thing when there is back tracking involved. The total distance traveled is computed with the speed, the change in position is computed with the velocity. When there is no change of sign, it is a bit more natural, perhaps, to use the language of speed and distance.
|
||
|
||
:::
|
||
|
||
##### Example: Galileo's ball and ramp experiment
|
||
|
||
|
||
One of history's most famous experiments was performed by [Galileo](http://en.wikipedia.org/wiki/History_of_experiments) where he rolled balls down inclined ramps, like that in @fig-picture-of-galileo-ramp, making note of distance traveled with respect to time. As Galileo had no ultra-accurate measuring device, he needed to slow movement down by controlling the angle of the ramp. With this, he could measure units of distance per units of time.
|
||
|
||
Suppose that no matter what the incline was, Galileo observed that in units of the distance traveled in the first second that the distance traveled between subsequent seconds was $3$ times, then $5$ times, then $7$ times, ... @tbl-galileo-distance-time-delta summarizes.
|
||
|
||
::: {#tbl-galileo-distance-time-delta .striped .hover}
|
||
| t | distance | delta |
|
||
|:------|:--------|:---------|
|
||
| 0 | 0 | . |
|
||
| 1 | 1 | 1 |
|
||
| 2 | 4 | 3 |
|
||
| 3 | 9 | 5 |
|
||
| 4 | 16 | 7 |
|
||
| 5 | 25 | 9 |
|
||
|
||
Distance traveled by time $t$ with differences computed
|
||
:::
|
||
|
||
The graph of distance versus time in @fig-plot-of-distance-versus-time-galileo-simple-data is found by interpolating between the measured points.
|
||
|
||
::: {#fig-plot-of-distance-versus-time-galileo-simple-data}
|
||
```{julia}
|
||
ts = [0,1,2,3,4, 5]
|
||
xs = [0,1,4,9,16,25]
|
||
plot(ts, xs)
|
||
```
|
||
|
||
Plot of distance traveled versus time
|
||
:::
|
||
|
||
The graph looks almost quadratic. What would the following questions have yielded?
|
||
|
||
|
||
* What is the average speed between $0$ and $3$?
|
||
|
||
|
||
```{julia}
|
||
(9-0) / (3-0) # (xs[4] - xs[1]) / (ts[4] - ts[1])
|
||
```
|
||
|
||
* What is the average speed between $2$ and $3$?
|
||
|
||
|
||
```{julia}
|
||
(9-4) / (3-2) # (xs[4] - xs[3]) / (ts[4] - ts[3])
|
||
```
|
||
|
||
From the graph, we can tell that the slope of the line connecting $(2,4)$ and $(3,9)$ will be greater than that connecting $(0,0)$ and $(3,9)$. In fact, given the shape of the graph (concave up), the line connecting $(0,0)$ with any point will have a slope less than or equal to any of the line segments.
|
||
|
||
|
||
The average speed between $k$ and $k+1$ for this graph is:
|
||
|
||
|
||
```{julia}
|
||
xs[2]-xs[1], xs[3] - xs[2], xs[4] - xs[3], xs[5] - xs[4]
|
||
```
|
||
|
||
We see it increments by $2$. The acceleration is the rate of change of speed. We see the rate of change of speed is constant, as the speed increments by $2$ each time unit.
|
||
|
||
|
||
Based on this---and given Galileo's insight---it appears the acceleration for a falling body subject to gravity will be **constant** and the position as a function of time will be quadratic.
|
||
|
||
|
||
## The slope of the secant line
|
||
|
||
|
||
In the above examples, we see that the average speed is computed using the slope formula. This can be generalized for any univariate function $f(x)$:
|
||
|
||
::: {.definition title="Average rate of change"}
|
||
|
||
For a continuous function $f$ the average rate of change between $a$ and $b$ is defined by
|
||
|
||
$$
|
||
\frac{f(b) - f(a)}{b - a}.
|
||
$$
|
||
|
||
It is typical to express this as $\Delta y/ \Delta x$, where $\Delta$ means "change".
|
||
:::
|
||
|
||
|
||
Geometrically, the average rate of change is the slope of the line connecting the points $(a, f(a))$ and $(b, f(b))$. This line is called a [secant](http://en.wikipedia.org/wiki/Secant_line) line---a line intersecting two specified points on a curve.
|
||
|
||
|
||
## The slope of the tangent line
|
||
|
||
Before continuing, we re-parameterize with $a=c$ and $b=c+h$ thinking of $c$ as fixed, and $h$ as varying. The secant-line-slope formula becomes:
|
||
|
||
|
||
$$
|
||
\text{slope (of a secant line)} = \frac{f(c+h) - f(c)}{h}.
|
||
$$
|
||
|
||
|
||
|
||
The slope of the secant line represents the average rate of change over a given period, $h$. What if this rate is so variable, that it makes sense to take smaller and smaller periods $h$? In fact, what if $h$ goes to $0$?
|
||
|
||
::: {#fig-secant-line-tangent-line-animation}
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
#| cache: true
|
||
gr()
|
||
function secant_line_tangent_line_graph(n)
|
||
f(x) = sin(x)
|
||
c = pi/3
|
||
h = 2.0^(-n) * pi/4
|
||
m = (f(c+h) - f(c))/h
|
||
|
||
xs = range(0, stop=pi, length=50)
|
||
fig_size=(800, 600)
|
||
plt = plot(;
|
||
xaxis=([], false),
|
||
yaxis=([], false),
|
||
framestyle=:origin,
|
||
legend=false,
|
||
|
||
ylims=(-.1,1.5)
|
||
)
|
||
|
||
plot!(f, 0, pi/2; line=(:black, 2))
|
||
plot!(f, pi/2, pi/2 + pi/5; line=(:black, 2, 1/4))
|
||
plot!(f, pi/2 + pi/5, pi; line=(:black, 2))
|
||
|
||
|
||
plot!(0.1 .+ [0,0],[-.1, 1.5]; line=(:gray,1), arrow=true, side=:head)
|
||
plot!([-0.2, 3.4], [.1, .1]; line=(:gray, 1), arrow=true, side=:head)
|
||
|
||
plot!(plt, xs, f(c) .+ cos(c)*(xs .- c), color=:orange)
|
||
plot!(plt, xs, f(c) .+ m*(xs .- c), color=:black)
|
||
scatter!(plt, [c,c+h], [f(c), f(c+h)], color=:orange, markersize=5)
|
||
|
||
plot!(plt, [c, c+h, c+h], [f(c), f(c), f(c+h)], color=:gray30)
|
||
|
||
annotate!(plt, [(c+h/2, f(c), text(L"h", :top)),
|
||
(c + h + .05, (f(c) + f(c + h))/2, text(L"f(c+h) - f(c)", :left)),
|
||
|
||
|
||
])
|
||
|
||
plt
|
||
end
|
||
caption = ""
|
||
n = 6
|
||
anim = @animate for i=0:n
|
||
secant_line_tangent_line_graph(i)
|
||
end
|
||
|
||
imgfile = tempname() * ".gif"
|
||
gif(anim, imgfile, fps = 1)
|
||
|
||
plotly()
|
||
ImageFile(imgfile, caption)
|
||
```
|
||
|
||
The slope of each secant line represents the *average* rate of change between $c$ and $c+h$. As $h$ goes towards $0$, we recover the slope of the tangent line, which represents the *instantatneous* rate of change.
|
||
:::
|
||
|
||
|
||
@fig-secant-line-tangent-line-animation suggests that for this function and at the point $x=c$ the slopes of the secant line converge to the slope of a "tangent" line. That is this limit exists:
|
||
|
||
|
||
$$
|
||
\lim_{h \rightarrow 0} \frac{f(c+h) - f(c)}{h}.
|
||
$$
|
||
|
||
With this, we define
|
||
|
||
::: {.definition title="Tangent line at c"}
|
||
|
||
When the following limit exists, the tangent line to the graph of $f(x)$ at $x=c$ is the line through the point $(c, f(c))$ with slope:
|
||
|
||
$$
|
||
m = \lim_{h \rightarrow 0} \frac{f(c+h) - f(c)}{h}.
|
||
$$
|
||
|
||
In point-slope form, the line is described by: $y = f(c) + m \cdot (x-c)$.
|
||
|
||
Later we will write $f'(c)$ for $m$.
|
||
:::
|
||
|
||
|
||
Informally, the tangent line is the line through the point that best approximates the function, as in @fig-tangent_line_approx_graph. The tangent line is not just a line that intersects the graph in one point, nor does it need only intersect the line in just one point.
|
||
|
||
::: {#fig-tangent_line_approx_graph}
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
let
|
||
gr()
|
||
function make_plot(Δ)
|
||
f(x) = 1 + sin(x-c)
|
||
df(x) = cos(x-c)
|
||
plt = plot(;
|
||
#xaxis=([], false),
|
||
yaxis=([], false),
|
||
aspect_ratio=:equal,
|
||
legend=false,
|
||
|
||
)
|
||
|
||
c = 1
|
||
xticks!([c-Δ, c, c+Δ], [latexstring("c-$Δ"), L"c", latexstring("c-$Δ")])
|
||
y₀ = f(c) - 2/3 * Δ
|
||
tl(x) = f(c) + df(c) * (x-c)
|
||
|
||
plot!(f, c - Δ, c + Δ; line=(:black, 2))
|
||
plot!(tl, c - Δ, c + Δ; line=(:red, 2))
|
||
plot!([c,c], [tl(c-Δ), f(c)]; line=(:gray, :dash, 1))
|
||
#plot!([c-1.1*Δ, c+1.1*Δ], y₀ .+ [0,0]; line=(:gray, 1), arrow=true)
|
||
|
||
|
||
current()
|
||
end
|
||
|
||
ps = make_plot.((1.5, 1.0, 0.5, 0.1))
|
||
|
||
plotly()
|
||
plot(ps...)
|
||
end
|
||
```
|
||
|
||
Illustration that the tangent line is the best linear approximation to $f(x)$ *near* $c$.
|
||
:::
|
||
|
||
|
||
|
||
|
||
:::{.callout-note}
|
||
## Note
|
||
This last point was certainly not obvious at first. [Barrow](http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf), who had Newton as a pupil, and was the first to sketch a proof of part of the Fundamental Theorem of Calculus, understood a tangent line to be a line that intersects a curve at only one point.
|
||
|
||
:::
|
||
|
||
##### Example
|
||
|
||
|
||
What is the slope of the tangent line to $f(x) = \sin(x)$ at $c=0$?
|
||
|
||
|
||
We need to compute the limit $(\sin(c+h) - \sin(c))/h$ which is the limit as $h$ goes to $0$ of $\sin(h)/h.$ We know this to be $1$. See @fig-plot-sin-tangent-line for a graph of the tangent line to the function.
|
||
|
||
::: {#fig-plot-sin-tangent-line}
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
f(x) = sin(x)
|
||
c = 0
|
||
tl(x) = f(c) + 1 * (x - c)
|
||
plot(f, -pi/2, pi/2; label="sin(x)")
|
||
plot!(tl; label="Tangent line")
|
||
```
|
||
|
||
Plot of $f(x) = \sin(x)$ with its tangent line ($y = 0 + 1 \cdot (x-0)$) at the origin
|
||
:::
|
||
|
||
## The derivative
|
||
|
||
|
||
The limit of the slopes of the secant lines for a fixed $c$ gives an operation: for each $c$ in the domain of $f$ when the limit exists
|
||
associate the slope of the tangent line.
|
||
|
||
::: {.definition title="The derivative"}
|
||
|
||
For each $x$ in the domain of $f(x)$, let $f'(x)$ define a function whose domain is all $x$ for which the following limit exists and whose value is given by:
|
||
|
||
$$
|
||
f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}.
|
||
$$
|
||
|
||
The function $f'(x)$ is called the *derivative* of $f(x)$.
|
||
:::
|
||
|
||
There are many notations for the derivative, mostly we use the "prime" notation in the definition, but at times it is more convenient to write only the expression defining the rule of the function. In that case, we use this notation for the derivative $[\text{expression}]'$.
|
||
|
||
|
||
### Some basic derivatives
|
||
|
||
* **The constant rule**. If $f(x) = c$, some constant, then the graph has zero slope, as $f(x+h) - f(x) = 0$ for all $x$ and $h$. That is the derivative is constantly $0$, or $[c]' = 0$.
|
||
|
||
* **The power rule**. What is the derivative of the monomial $f(x) = x^n$? We need to look at $(x+h)^n - x^n$ for positive, integer-value $n$. Let's look at a case, $n=5$
|
||
|
||
|
||
```{julia}
|
||
@syms x::real h::real
|
||
n = 5
|
||
ex = expand((x+h)^n - x^n)
|
||
```
|
||
|
||
All terms have an `h` in them, so we cancel it out:
|
||
|
||
|
||
```{julia}
|
||
cancel(ex/h, h)
|
||
```
|
||
|
||
We see the lone term `5x^4` without an $h$, so as we let $h$ go to $0$, this will be the limit. That is, $f'(x) = 5x^4$.
|
||
|
||
|
||
For integer-valued, positive, $n$, the binomial theorem gives an expansion $(x+h)^n = x^n + nx^{n-1}\cdot h^1 + n\cdot(n-1)x^{n-2}\cdot h^2 + \cdots$. Subtracting $x^n$ then dividing by $h$ leaves just the term $nx^{n-1}$ without a power of $h$, so the limit, in general, is just this term. That is:
|
||
|
||
|
||
$$
|
||
[x^n]' = nx^{n-1}.
|
||
$$
|
||
|
||
It isn't a special case, but when $n=0$, we also have the above formula applies, as $x^0$ is the constant $1$. We will see later that in general, the power rule applies for any $n$ where $x^n$ is defined.
|
||
|
||
|
||
* What is the derivative of $f(x) = \sin(x)$? We know that $f'(0)= 1$ by the earlier example with $(\sin(0+h)-\sin(0))/h = \sin(h)/h$, here we solve in general.
|
||
|
||
|
||
We need to consider the difference $\sin(x+h) - \sin(x)$:
|
||
|
||
|
||
```{julia}
|
||
sympy.expand_trig(sin(x+h) - sin(x)) # expand_trig is not exposed in `SymPy`
|
||
```
|
||
|
||
We could then rearrange the secant line slope formula to become:
|
||
|
||
|
||
$$
|
||
\cos(x) \cdot \frac{\sin(h)}{h} + \sin(x) \cdot \frac{\cos(h) - 1}{h}.
|
||
$$
|
||
|
||
We then take a limit as $h \rightarrow 0$. If the answer isn't clear, we can let `SymPy` do this work:
|
||
|
||
|
||
```{julia}
|
||
limit((sin(x+h) - sin(x))/ h, h => 0)
|
||
```
|
||
|
||
From the formula $[\sin(x)]' = \cos(x)$ we can easily get the *slope* of the tangent line to $f(x) = \sin(x)$ at $x=0$ by simply evaluating $\cos(0) = 1$.
|
||
|
||
|
||
* Let's see what the derivative of $\ln(x) = \log(x)$ is (using base $e$ for $\log$ unless otherwise indicated). We have
|
||
|
||
|
||
$$
|
||
\frac{\log(x+h) - \log(x)}{h} = \frac{1}{h}\log(\frac{x+h}{x}) = \log((1+h/x)^{1/h}).
|
||
$$
|
||
|
||
As noted earlier, Cauchy saw the limit as $u$ goes to $0$ of $f(u) = (1 + u)^{1/u}$ is $e$. Re-expressing the above we can get $1/x \cdot \log(f(h/x))$. The limit as $h$ goes to $0$ of this is found from the composition rules for limits: as $\lim_{h \rightarrow 0} f(h/x) = e$, and since $\log(x)$ is continuous at $e$ we get this expression has a limit of $1/x$.
|
||
|
||
|
||
We verify through:
|
||
|
||
|
||
```{julia}
|
||
limit((log(x+h) - log(x))/h, h => 0)
|
||
```
|
||
|
||
* The derivative of $f(x) = e^x$ can also be done from a limit. We have
|
||
|
||
|
||
$$
|
||
\frac{e^{x+h} - e^x}{h} = \frac{e^x \cdot(e^h -1)}{h}.
|
||
$$
|
||
|
||
Earlier, we saw that $\lim_{h \rightarrow 0}(e^h - 1)/h = 1$. With this, we get $[e^x]' = e^x$, that is it is a function satisfying $f'(x)=f(x)$.
|
||
|
||
|
||
---
|
||
|
||
|
||
There are several [notations](http://en.wikipedia.org/wiki/Notation_for_differentiation) for derivatives. Some are historical, some just add flexibility. We use the prime notation of Lagrange: $f'(x)$, $u'$ and $[\text{expr}]'$, where the first emphasizes that the derivative is a function with a value at $x$, the second emphasizes the derivative operates on functions, the last emphasizes that we are taking the derivative of some expression.
|
||
|
||
|
||
Some other notations include:
|
||
|
||
|
||
* The Leibniz notation which uses the infinitesimals, $dy/dx$, to relate to $\Delta y/\Delta x$. This notation is very common, and especially useful when more than one variable is involved. `SymPy` uses Leibniz notation in some of its output, expressing somethings such as:
|
||
|
||
|
||
$$
|
||
f'(x) = \frac{d}{d\xi}(f(\xi)) \big|_{\xi=x}.
|
||
$$
|
||
|
||
The notation--$\big|$---on the right-hand side separates the tasks of finding the derivative and evaluating the derivative at a specific value.
|
||
|
||
|
||
* Euler used `D` for the operator `D(f)`. This was initially used by [Argobast](http://jeff560.tripod.com/calculus.html). The notation `D(f)(c)` would be needed to evaluate the derivative at a point.
|
||
|
||
* Newton used a "dot" above the variable, $\dot{x}(t)$, which is still widely used in physics to indicate a derivative in time. This indicates first taking the derivative and then plugging in $t$.
|
||
|
||
* The notation $[expr]'(c)$ or $[expr]'\big|_{x=c}$would similarly mean, take the derivative of the expression and **then** evaluate at $c$.
|
||
|
||
|
||
## Rules of derivatives
|
||
|
||
|
||
We could proceed in a similar manner---using limits to find other derivatives, but let's not. If we have a function $f(x) = x^5 \sin(x)$, it would be nice to leverage our previous work on the derivatives of $f(x) =x^5$ and $g(x) = \sin(x)$, rather than derive an answer from scratch.
|
||
|
||
|
||
As with limits and continuity, it proves very useful to consider rules that make the process of finding derivatives of combinations of functions a matter of combining derivatives of the individual functions in some manner.
|
||
|
||
|
||
We already have one such rule:
|
||
|
||
|
||
::: {.relationship title="Power rule"}
|
||
|
||
For integer $n \geq 0$ the power rule is:
|
||
|
||
|
||
$$
|
||
[x^n]' = n x^{n-1}.
|
||
$$
|
||
|
||
The power rule will be shown true for all real exponents.
|
||
:::
|
||
|
||
|
||
We now discuss rules which express derivatives of compound expressions in terms of derivatives of different pieces of an expression.
|
||
|
||
|
||
Let's consider $k(x) = a\cdot f(x) + b\cdot g(x)$, what is its derivative? That is, in terms of $f$, $g$ and their derivatives, can we express $k'(x)$?
|
||
|
||
|
||
We can rearrange $(k(x+h) - k(x))$ as follows:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
(a\cdot f(x+h) + b\cdot g(x+h)) - (a\cdot f(x) + b \cdot g(x)) &=\\
|
||
\quad a\cdot (f(x+h) - f(x)) + b \cdot (g(x+h) - g(x)). &
|
||
\end{align*}
|
||
$$
|
||
|
||
Dividing by $h$, we see that this becomes
|
||
|
||
|
||
$$
|
||
a\cdot \frac{f(x+h) - f(x)}{h} + b \cdot \frac{g(x+h) - g(x)}{h} \rightarrow a\cdot f'(x) + b\cdot g'(x).
|
||
$$
|
||
|
||
That is $[a\cdot f(x) + b \cdot g(x)]' = a\cdot f'(x) + b\cdot g'(x)$.
|
||
|
||
|
||
::: {.relationship title="Constant multiple and sum rule"}
|
||
The sum rule applies to constant multiple and sums of functions:
|
||
|
||
$$
|
||
[a f(x) + b g(x)]' = a f'(x) + b g'(x)
|
||
$$
|
||
|
||
This holds two rules: the derivative of a constant times a function is the constant times the derivative of the function; and the derivative of a sum of functions is the sum of the derivative of the functions.
|
||
|
||
:::
|
||
|
||
|
||
This example shows a useful template:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[2x^2 - \frac{x}{3} + 3e^x]' & = 2[\square]' - \frac{[\square]'}{3} + 3[\square]'\\
|
||
&= 2[x^2]' - \frac{[x]'}{3} + 3[e^x]'\\
|
||
&= 2(2x) - \frac{1}{3} + 3e^x\\
|
||
&= 4x - \frac{1}{3} + 3e^x
|
||
\end{align*}
|
||
$$
|
||
|
||
----
|
||
|
||
Other rules can be similarly derived. We simply state the rules for products and quotients.
|
||
|
||
::: {.relationship title="Product rule"}
|
||
The derivative of a product of functions is given by:
|
||
|
||
$$
|
||
[f(x) \cdot g(x)]' = f'(x)\cdot g(x) + f(x) \cdot g'(x)
|
||
$$
|
||
|
||
|
||
A common shorthand is $[uv]' = u'v + uv'$.
|
||
:::
|
||
|
||
|
||
This example shows a useful template for the product rule:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[(x^2+1)\cdot e^x]' &= [\square]' \cdot (\square) + (\square) \cdot [\square]'\\
|
||
&= [x^2 + 1]' \cdot (e^x) + (x^2+1) \cdot [e^x]'\\
|
||
&= (2x)\cdot e^x + (x^2+1)\cdot e^x
|
||
\end{align*}
|
||
$$
|
||
|
||
----
|
||
|
||
The derivative of $f(x) = u(x)/v(x)$---a ratio of functions---can be similarly computed.
|
||
|
||
::: {.relationship title="Quotient rule"}
|
||
The derivative of a ratio of functions can be computed by the quotient rule:
|
||
|
||
$$
|
||
\left[\frac{f(x)}{g(x)}\right]' = \frac{f'(x)\cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}.
|
||
$$
|
||
|
||
This is often written as $[u/v]' = (u'v - uv')/v^2$.
|
||
:::
|
||
|
||
|
||
|
||
This example shows a useful template for the quotient rule:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[\frac{x^2+1}{e^x}]' &= \frac{[\square]' \cdot (\square) - (\square) \cdot [\square]'}{(\square)^2}\\
|
||
&= \frac{[x^2 + 1]' \cdot (e^x) - (x^2+1) \cdot [e^x]'}{(e^x)^2}\\
|
||
&= \frac{(2x)\cdot e^x - (x^2+1)\cdot e^x}{e^{2x}}
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
##### Examples
|
||
|
||
|
||
Compute the derivative of $f(x) = (1 + \sin(x)) + (1 + x^2)$.
|
||
|
||
|
||
As written we can identify $f(x) = u(x) + v(x)$ with $u=(1 + \sin(x))$, $v=(1 + x^2)$. The sum rule immediately applies to give:
|
||
|
||
|
||
$$
|
||
f'(x) = (\cos(x)) + (2x).
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Compute the derivative of $f(x) = (1 + \sin(x)) \cdot (1 + x^2)$.
|
||
|
||
|
||
The same $u$ and $v$ my be identified. The product rule readily applies to yield:
|
||
|
||
|
||
$$
|
||
f'(x) = u'v + uv' = \cos(x) \cdot (1 + x^2) + (1 + \sin(x)) \cdot (2x).
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Compute the derivative of $f(x) = (1 + \sin(x)) / (1 + x^2)$.
|
||
|
||
|
||
The same $u$ and $v$ my be identified. The quotient rule readily applies to yield:
|
||
|
||
|
||
$$
|
||
f'(x) = \frac{u'v - uv'}{v^2} = \frac{\cos(x) \cdot (1 + x^2) - (1 + \sin(x)) \cdot (2x)}{(1+x^2)^2}.
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Compute the derivative of $f(x) = (x-1) \cdot (x-2)$.
|
||
|
||
|
||
This can be done using the product rule *or* by expanding the polynomial and using the power and sum rule. As this polynomial is easy to expand, we do both and compare:
|
||
|
||
|
||
$$
|
||
[(x-1)(x-2)]' = [x^2 - 3x + 2]' = 2x -3.
|
||
$$
|
||
|
||
Whereas the product rule gives:
|
||
|
||
|
||
$$
|
||
[(x-1)(x-2)]' = 1\cdot (x-2) + (x-1)\cdot 1 = 2x - 3.
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)$.
|
||
|
||
|
||
We could expand this, as above, but without computer assistance the potential for error is high. Instead we will use the product rule on the product of $5$ terms.
|
||
|
||
|
||
Let's first treat the case of $3$ products:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[u\cdot v\cdot w]' &=[ u \cdot (vw)]'\\
|
||
&= u' (vw) + u [vw]'\\
|
||
&= u'(vw) + u[v' w + v w'] \\
|
||
&=u' vw + u v' w + uvw'.
|
||
\end{align*}
|
||
$$
|
||
|
||
This pattern generalizes, clearly, to:
|
||
|
||
|
||
$$
|
||
[f_1\cdot f_2 \cdots f_n]' = f_1' f_2 \cdots f_n + f_1 \cdot f_2' \cdot f_3 \cdots f_n + \dots +
|
||
f_1 \cdots f_{n-1} \cdot f_n'.
|
||
$$
|
||
|
||
There are $n$ terms, each where one of the $f_i$s have a derivative. Were we to multiply top and bottom by $f_i$, we would get each term looks like: $f \cdot f_i'/f_i$.
|
||
|
||
|
||
With this, we can proceed. Each term $x-i$ has derivative $1$, so the answer to $f'(x)$, with $f$ as above, is
|
||
$$
|
||
\begin{align*}
|
||
f'(x) &= f(x)/(x-1) + f(x)/(x-2) + f(x)/(x-3)\\
|
||
&+ f(x)/(x-4) + f(x)/(x-5),
|
||
\end{align*}
|
||
$$
|
||
|
||
That is
|
||
$$
|
||
\begin{align*}
|
||
f'(x) &= (x-2)(x-3)(x-4)(x-5) + (x-1)(x-3)(x-4)(x-5)\\
|
||
&+ (x-1)(x-2)(x-4)(x-5) + (x-1)(x-2)(x-3)(x-5) \\
|
||
&+ (x-1)(x-2)(x-3)(x-4).
|
||
\end{align*}
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $x\sin(x)$ evaluated at $\pi$.
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[x\sin(x)]'\big|_{x=\pi} &= (1\sin(x) + x\cos(x))\big|_{x=\pi} \\
|
||
&= (\sin(\pi) + \pi \cdot \cos(\pi)) \\
|
||
&= -\pi.
|
||
\end{align*}
|
||
$$
|
||
|
||
### Chain rule
|
||
|
||
|
||
Finally, the derivative of a composition of functions can be computed using pieces of each function. This gives a rule called the *chain rule*. Before deriving, let's give a slight motivation through an example.
|
||
|
||
Consider the output of a factory for some widget. It depends on two steps: an initial manufacturing step and a finishing step. The number of employees is important in how much is initially manufactured. Suppose $x$ is the number of employees and $g(x)$ is the amount initially manufactured. Adding more employees increases the amount made by the made-up rule $g(x) = \sqrt{x}$. The finishing step depends on how much is made by the employees. If $y$ is the amount made, then $f(y)$ is the number of widgets finished. Suppose for some reason that $f(y) = y^2.$
|
||
|
||
|
||
How many widgets are made as a function of employees? The composition $u(x) = f(g(x))$ would provide that. Changes in the initial manufacturing step lead to changes in how much is initially made; changes in the initial amount made leads to changes in the finished products. Each change contributes to the overall change.
|
||
|
||
|
||
What is the effect of adding employees on the rate of output of widgets? In this specific case we know the answer, as $(f \circ g)(x) = x$, so the answer is just the rate is $1$.
|
||
|
||
|
||
In general, we want to express $\Delta f / \Delta x$ in a form so that we can take a limit.
|
||
|
||
|
||
But what do we know? We know $\Delta g / \Delta x$ and $\Delta f/\Delta y$. Using $y=g(x)$, this suggests that we might have luck with the right side of this equation:
|
||
|
||
|
||
$$
|
||
\frac{\Delta f}{\Delta x} = \frac{\Delta f}{\Delta y} \cdot \frac{\Delta y}{\Delta x}.
|
||
$$
|
||
|
||
Interpreting this, we get the *average* rate of change in the composition can be thought of as a product: The *average* rate of change of the initial step ($\Delta y/ \Delta x$) times the *average* rate of the change of the second step evaluated not at $x$, but at $y$, $\Delta f/ \Delta y$.
|
||
|
||
|
||
Re-expressing using derivative notation with $h$ would be:
|
||
|
||
|
||
$$
|
||
\frac{f(g(x+h)) - f(g(x))}{h} = \frac{f(g(x+h)) - f(g(x))}{g(x+h) - g(x)} \cdot \frac{g(x+h) - g(x)}{h}.
|
||
$$
|
||
|
||
The left-hand side will converge to the derivative of $u(x)$ or $[f(g(x))]'$.
|
||
|
||
|
||
The right-most part of the right-hand side would have a limit $g'(x)$, were we to let $h$ go to $0$.
|
||
|
||
|
||
It isn't obvious, but the left part of the right-hand side has the limit $f'(g(x))$. This would be clear if *only* $g(x+h) = g(x) + h$, for then the expression would be exactly the limit expression with $c=g(x)$. But, alas, except to some hopeful students and some special cases, it is definitely not the case in general that $g(x+h) = g(x) + h$---that right parentheses actually means something. However, it is *nearly* the case that $g(x+h) = g(x) + kh$ for some $k$ and this can be used to formulate a proof, like the one sketched below.
|
||
|
||
|
||
Combined, we would end up with the chain rule
|
||
|
||
::: {.relationship title="Chain rule"}
|
||
The *chain rule* is used to find derivatives of compositions:
|
||
|
||
|
||
$$
|
||
[f(g(x))]' = f'(g(x)) \cdot g'(x).
|
||
$$
|
||
|
||
That is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
|
||
:::
|
||
|
||
|
||
To see that the chain rule works in our specific case, we assume the general power rule that $[x^n]' = n x^{n-1}$ (for all real $n$) to get:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(x) &= x^2 & g(x) &= \sqrt{x}\\
|
||
f'(\square) &= 2(\square) & g'(x) &= \frac{1}{2}x^{-1/2}
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
We use $\square$ for the argument of $f'$ to emphasize that $g(x)$ is the needed value, not just $x$:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[(\sqrt{x})^2]' &= [f(g(x)]'\\
|
||
&= f'(g(x)) \cdot g'(x) \\
|
||
&= 2(\sqrt{x}) \cdot \frac{1}{2}x^{-1/2}\\
|
||
&= \frac{2\sqrt{x}}{2\sqrt{x}}\\
|
||
&=1
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
This is the same as the derivative of $x$ found by first evaluating the composition. For this problem, the chain rule is not necessary, but typically it is the most important rule to fully differentiate a function.
|
||
|
||
|
||
##### Examples
|
||
|
||
|
||
Find the derivative of $f(x) = \sqrt{1 - x^2}$. We identify the composition of $\sqrt{x}$ and $(1-x^2)$. We set the functions and their derivatives into a pattern to emphasize the pieces in the chain-rule formula:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(x) &=\sqrt{x} = x^{1/2} & g(x) &= 1 - x^2 \\
|
||
f'(\square) &=(1/2)(\square)^{-1/2} & g'(x) &= -2x
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
Then:
|
||
|
||
|
||
$$
|
||
[f(g(x))]' = (1/2)(1-x^2)^{-1/2} \cdot (-2x).
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $\log(2 + \sin(x))$. This is a composition $\log(x)$---with derivative $1/x$ and $2 + \sin(x)$---with derivative $\cos(x)$. We get $(1/(2 + \sin(x))) \cos(x)$.
|
||
|
||
|
||
In general,
|
||
|
||
|
||
$$
|
||
[\log(f(x))]' = \frac{f'(x)}{f(x)}.
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $e^{f(x)}$. The inner function has derivative $f'(x)$, the outer function has derivative $e^x$ (the same as the outer function itself). We get for a derivative
|
||
|
||
|
||
$$
|
||
[e^{f(x)}]' = e^{f(x)} \cdot f'(x).
|
||
$$
|
||
|
||
This is a useful rule to remember for expressions involving exponentials.
|
||
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $\sin(x)\cos(2x)$ at $x=\pi$.
|
||
|
||
$$
|
||
\begin{align*}
|
||
[\sin(x)\cos(2x)]'\big|_{x=\pi} &=(\cos(x)\cos(2x) + \sin(x)(-\sin(2x)\cdot 2))\big|_{x=\pi} \\
|
||
& =((-1)(1) + (0)(-0)(2)) = -1.
|
||
\end{align*}
|
||
$$
|
||
|
||
##### Proof of the Chain Rule
|
||
|
||
|
||
A function is *differentiable* at $a$ if the following limit exists $\lim_{h \rightarrow 0}(f(a+h)-f(a))/h$.
|
||
This is reexpressed as $f$ is differentitable at $a$ is there exists $\epsilon_f(h) \rightarrow 0$ such that:
|
||
|
||
$$
|
||
f(a+h) - f(a) - f'(a)h = \epsilon_f(h) h.
|
||
$$
|
||
|
||
|
||
As $g$ is differentiable, we have:
|
||
|
||
|
||
$$
|
||
g(a+h) = g(a) + g'(a)h + \epsilon_g(h) h = g(a) + h',
|
||
$$
|
||
|
||
Where $h' = g'(a)h + \epsilon_g(h)h \rightarrow 0$ as $h \rightarrow 0$ will be used to simplify the following:
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(g(a+h)) - f(g(a)) &=
|
||
f(g(a) + g'(a)h + \epsilon_g(h)h) - f(g(a)) \\
|
||
&= \textcolor{red}{f(g(a))} + f'(g(a)) (g'(a)h + \epsilon_g(h)h) + \epsilon_f(h')(h') \\
|
||
&\qquad- \textcolor{red}{f(g(a))}\\
|
||
&= f'(g(a)) g'(a)h + \textcolor{blue}{f'(g(a))(\epsilon_g(h)h) + \epsilon_f(h')(h')}.
|
||
\end{align*}
|
||
$$
|
||
|
||
Rearranging:
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(g(a+h)) &- f(g(a)) - f'(g(a)) g'(a) h\\
|
||
&= \textcolor{blue}{f'(g(a))\epsilon_g(h)h + \epsilon_f(h')(h')}\\
|
||
&=(f'(g(a)) \epsilon_g(h) + \epsilon_f(h') (g'(a) + \epsilon_g(h)))h \\
|
||
&=\epsilon(h)h,
|
||
\end{align*}
|
||
$$
|
||
|
||
where $\epsilon(h)$ combines the above terms which go to zero as $h\rightarrow 0$ into one. This is the alternative definition of differentiable, showing $(f\circ g)'(a) = f'(g(a)) g'(a)$ when $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$.
|
||
|
||
|
||
##### The "chain" rule
|
||
|
||
|
||
The chain rule name could also be simply the "composition rule," as that is the operation the rule works for. However, in practice, there are usually *multiple* compositions, and the "chain" rule is used to chain together the different pieces. To get a sense, consider a triple composition $u(v(w(x)))$. This will have derivative:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[u(v(w(x)))]' &= u'(v(w(x))) \cdot [v(w(x))]' \\
|
||
&= u'(v(w(x))) \cdot v'(w(x)) \cdot w'(x)
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
The answer can be viewed as a repeated peeling off of the outer function, a view with immediate application to many compositions. To see that in action with an expression, consider this derivative problem, shown in steps:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[\sin(e^{\cos(x^2-x)})]'
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot [e^{\cos(x^2-x)}]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot [\cos(x^2-x)]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot [x^2-x]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot (2x-1)\\
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
##### More examples of differentiation
|
||
|
||
|
||
Find the derivative of $x^5 \cdot \sin(x)$.
|
||
|
||
|
||
This is a product of functions, using $[u\cdot v]' = u'v + uv'$ we get:
|
||
|
||
|
||
$$
|
||
5x^4 \cdot \sin(x) + x^5 \cdot \cos(x)
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $x^5 / \sin(x)$.
|
||
|
||
|
||
This is a quotient of functions. Using $[u/v]' = (u'v - uv')/v^2$ we get
|
||
|
||
|
||
$$
|
||
(5x^4 \cdot \sin(x) - x^5 \cdot \cos(x)) / (\sin(x))^2.
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $\sin(x^5)$. This is a composition of functions $u(v(x))$ with $v(x) = x^5$. The chain rule says find the derivative of $u$ ($\cos(x)$) and evaluate at $v(x)$ ($\cos(x^5)$) then multiply by the derivative of $v$:
|
||
|
||
|
||
$$
|
||
\cos(x^5) \cdot 5x^4.
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Similarly, but differently, find the derivative of $\sin(x)^5$. Now $v(x) = \sin(x)$, so the derivative of $u(x)$ ($5x^4$) evaluated at $v(x)$ is $5(\sin(x))^4$ so multiplying by $v'$ gives:
|
||
|
||
|
||
$$
|
||
5(\sin(x))^4 \cdot \cos(x)
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
We can verify these with `SymPy`.
|
||
|
||
|
||
```{julia}
|
||
diff(x^5 * sin(x), x)
|
||
```
|
||
|
||
```{julia}
|
||
diff(x^5/sin(x), x)
|
||
```
|
||
|
||
```{julia}
|
||
diff(sin(x^5), x)
|
||
```
|
||
|
||
and finally,
|
||
|
||
|
||
```{julia}
|
||
diff(sin(x)^5, x)
|
||
```
|
||
|
||
---
|
||
|
||
Find the derivative of $f(x) = x^3 (1-x)^2$ using either the power rule or the sum rule.
|
||
|
||
|
||
The power rule expresses $f=u\cdot v$. With $u(x)=x^3$ and $v(x)=(1-x)^2$ we get:
|
||
|
||
|
||
$$
|
||
u'(x) = 3x^2, \quad v'(x) = 2 \cdot (1-x)^1 \cdot (-1),
|
||
$$
|
||
|
||
the last by the chain rule. Combining with $u' v + u v'$ we get: $f'(x) = (3x^2)\cdot (1-x)^2 + x^3 \cdot (-2) \cdot (1-x)$.
|
||
|
||
|
||
Otherwise, the polynomial can be expanded to give $f(x)=x^5-2x^4+x^3$ which has derivative $f'(x) = 5x^4 - 8x^3 + 3x^2$.
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $f(x) = x \cdot e^{-x^2}$.
|
||
|
||
|
||
Using the product rule and then the chain rule, we have:
|
||
$$
|
||
\begin{align*}
|
||
f'(x) &= [x \cdot e^{-x^2}]'\\
|
||
&= [x]' \cdot e^{-x^2} + x \cdot [e^{-x^2}]'\\
|
||
&= 1 \cdot e^{-x^2} + x \cdot (e^{-x^2}) \cdot [-x^2]'\\
|
||
&= e^{-x^2} + x \cdot e^{-x^2} \cdot (-2x)\\
|
||
&= e^{-x^2} (1 - 2x^2).
|
||
\end{align*}
|
||
$$
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $f(x) = e^{-ax} \cdot \sin(x)$.
|
||
|
||
|
||
Using the product rule and then the chain rule, we have:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
f'(x) &= [e^{-ax} \cdot \sin(x)]'\\
|
||
&= [e^{-ax}]' \cdot \sin(x) + e^{-ax} \cdot [\sin(x)]'\\
|
||
&= e^{-ax} \cdot [-ax]' \cdot \sin(x) + e^{-ax} \cdot \cos(x)\\
|
||
&= e^{-ax} \cdot (-a) \cdot \sin(x) + e^{-ax} \cos(x)\\
|
||
&= e^{-ax}(\cos(x) - a\sin(x)).
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
---
|
||
|
||
|
||
Find the derivative of $e^{-x^2/2}$ at $x=1$.
|
||
|
||
|
||
$$
|
||
[e^{-x^2/2}]'\big|_{x=1} =
|
||
(e^{-x^2/2} \cdot \frac{-2x}{2}) \big|_{x=1} =
|
||
e^{-1/2} \cdot (-1) = -e^{-1/2}.
|
||
$$
|
||
|
||
##### Example: the general product rule
|
||
|
||
The general product rule: For any real $n$---not just integer values---we can re-express $x^n$ using $e$:
|
||
|
||
$$
|
||
x^n = e^{\log(x^n)} = e^{n \log(x)}.
|
||
$$
|
||
|
||
|
||
Now the chain rule can be applied:
|
||
|
||
|
||
$$
|
||
[x^n]' = [e^{n\log(x)}]' = e^{n\log(x)} \cdot (n \frac{1}{x}) = n x^n \cdot \frac{1}{x} = n x^{n-1}.
|
||
$$
|
||
|
||
|
||
##### Example: derivative of inverse functions
|
||
|
||
|
||
Suppose we knew that $\log(x)$ had derivative of $1/x$, but didn't know the derivative of $e^x$. From their inverse relation, we have: $x=\log(e^x)$, so taking derivatives of both sides would yield:
|
||
|
||
|
||
$$
|
||
1 = (\frac{1}{e^x}) \cdot [e^x]'.
|
||
$$
|
||
|
||
Or solving, $[e^x]' = e^x$. This is a general strategy to find the derivative of an *inverse* function in terms of the derivative of the function.
|
||
|
||
|
||
The graph of an inverse function is related to the graph of the function through the symmetry $y=x$.
|
||
|
||
|
||
For example, the graph of $e^x$ and $\log(x)$ have this symmetry, emphasized in @fig-plot-f-inverse-several-points-showing-derivative.
|
||
|
||
::: {#fig-plot-f-inverse-several-points-showing-derivative}
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
f(x) = exp(x)
|
||
f′(x) = exp(x)
|
||
f⁻¹(x) = log(x) # using Unicode typed with "f^\-^\1"
|
||
xs = range(0, 2, length=25)
|
||
ys = f.(xs)
|
||
plot(f, 0, 2, aspect_ratio=:equal, xlim=(0,8), ylim=(0,8), legend=false)
|
||
scatter!(xs, ys)
|
||
plot!(f⁻¹, extrema(ys)...)
|
||
scatter!(ys, xs, color=:blue)
|
||
plot!(identity, linestyle=:dot) # the line y=x
|
||
x₀, y₀ = xs[13], ys[13]
|
||
plot!([x₀, y₀],[y₀, x₀], linestyle=:dash)
|
||
ys′ = @. y₀ + f(x₀)*(xs - x₀)
|
||
plot!(xs, ys′, linestyle=:dash)
|
||
g(y) = 1/f′(f⁻¹(y))
|
||
xs′ = @. x₀ + g(y₀) * (ys - y₀)
|
||
plot!(ys, xs′, linestyle=:dash)
|
||
```
|
||
|
||
Plot of $f(x) = e^x$ with its inverse function showing the relationship between tangent lines
|
||
:::
|
||
|
||
The point $(1, e)$ on the graph of $e^x$ matches the point $(e, 1)$ on the graph of the inverse function, $\log(x)$. The slope of the tangent line at $x=1$ to $e^x$ is given by $e$ as well. What is the slope of the tangent line to $\log(x)$ at $x=e$?
|
||
|
||
|
||
As seen, the value can be computed, but how?
|
||
|
||
|
||
Finding the derivative of the inverse function can be achieved from the chain rule using the identify $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.
|
||
|
||
|
||
The chain rule applied to both sides, yields:
|
||
|
||
|
||
$$
|
||
1 = [f^{-1}]'(f(x)) \cdot f'(x)
|
||
$$
|
||
|
||
Solving, we see that $[f^{-1}]'(f(x)) = 1/f'(x)$. To emphasize the evaluation of the derivative of the inverse function at $f(x)$ we might write:
|
||
|
||
|
||
$$
|
||
\frac{d}{du} (f^{-1}(u)) \big|_{u=f(x)} = \frac{1}{f'(x)}
|
||
$$
|
||
|
||
So the reciprocal of the slope of the tangent line of $f$ at the mirror image point. In the above, we see if the slope of the tangent line at $(1,e)$ to $f$ is $e$, then the slope of the tangent line to $f^{-1}(x)$ at $(e,1)$ would be $1/e$.
|
||
|
||
|
||
#### Rules of derivatives and some sample functions
|
||
|
||
|
||
This table summarizes the rules of derivatives that allow derivatives of more complicated expressions to be computed with the derivatives of their pieces.
|
||
|
||
::: {#tbl-rules-of-derivatives .striped .hover}
|
||
| Name | Rule |
|
||
| --------------:| ---------------------------------------------------------------:|
|
||
| Power rule | $[x^n]' = n\cdot x^{n-1}$ |
|
||
| constant | $[cf(x)]' = c \cdot f'(x)$ |
|
||
| sum/difference | $[f(x) \pm g(x)]' = f'(x) \pm g'(x)$ |
|
||
| product | $[f(x) \cdot g(x)]' = f'(x)\cdot g(x) + f(x) \cdot g'(x)$ |
|
||
| quotient | $[f(x)/g(x)]' = (f'(x) \cdot g(x) - f(x) \cdot g'(x)) / g(x)^2$ |
|
||
| chain | $[f(g(x))]' = f'(g(x)) \cdot g'(x)$ |
|
||
|
||
: Table summarizing different rules for derivatives
|
||
:::
|
||
|
||
@tbl-derivatives-of-key-functions shows the derivative of a few key functions.
|
||
|
||
::: {#tbl-derivatives-of-key-functions .striped .hover}
|
||
|
||
| Function | Derivative |
|
||
| ----------------------:| ----------:|
|
||
| $x^n (\text{ all } n)$ | $nx^{n-1}$ |
|
||
| $e^x$ | $e^x$ |
|
||
| $\log(x)$ | $1/x$ |
|
||
| $\sin(x)$ | $\cos(x)$ |
|
||
| $\cos(x)$ | $-\sin(x)$ |
|
||
|
||
Derivatives of some functions
|
||
:::
|
||
|
||
## Higher-order derivatives
|
||
|
||
|
||
The derivative of a function is an operator, it takes a function and returns a new, derived, function. We could repeat this operation. The result is called a higher-order derivative. The Lagrange notation uses additional "primes" to indicate how many. So $f''(x)$ is the second derivative and $f'''(x)$ the third. For even higher orders, sometimes the notation is $f^{(n)}(x)$ to indicate an $n$th derivative.
|
||
|
||
|
||
##### Examples
|
||
|
||
|
||
Find the first $3$ derivatives of $f(x) = ax^3 + bx^2 + cx + d$.
|
||
|
||
|
||
Differentiating a polynomial is done with the sum rule, here we repeat three times:
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(x) &= ax^3 + bx^2 + cx + d\\
|
||
f'(x) &= 3ax^2 + 2bx + c \\
|
||
f''(x) &= 3\cdot 2 a x + 2b \\
|
||
f'''(x) &= 6a
|
||
\end{align*}
|
||
$$
|
||
|
||
We can see, the fourth derivative---and all higher order ones---would be identically $0$. This is part of a general phenomenon: an $n$th degree polynomial has only $n$ non-zero derivatives.
|
||
|
||
|
||
---
|
||
|
||
|
||
Find the first $5$ derivatives of $\sin(x)$.
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
f(x) &= \sin(x) \\
|
||
f'(x) &= \cos(x) \\
|
||
f''(x) &= -\sin(x) \\
|
||
f'''(x) &= -\cos(x) \\
|
||
f^{(4)} &= \sin(x) \\
|
||
f^{(5)} &= \cos(x)
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
We see the derivatives repeat themselves. (We also see alternative notation for higher order derivatives.)
|
||
|
||
|
||
---
|
||
|
||
|
||
Find the second derivative of $e^{-x^2}$.
|
||
|
||
|
||
We need the chain rule *and* the product rule:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
[e^{-x^2}]''
|
||
&= [e^{-x^2} \cdot (-2x)]' \\
|
||
&= \left(e^{-x^2} \cdot (-2x)\right) \cdot(-2x) + e^{-x^2} \cdot (-2) \\
|
||
&= e^{-x^2}(4x^2 - 2).
|
||
\end{align*}
|
||
$$
|
||
|
||
This can be verified:
|
||
|
||
|
||
```{julia}
|
||
diff(diff(exp(-x^2))) |> simplify
|
||
```
|
||
|
||
::: {.callout-note}
|
||
## higher order derivatives with `diff`
|
||
|
||
Having to iterate the use of `diff` is cumbersome. An alternate notation is either
|
||
|
||
* specifying the variable twice: `diff(ex, x, x)` or
|
||
* using a number after the variable: `diff(ex, x, 2)`:
|
||
|
||
For example,
|
||
|
||
```{julia}
|
||
diff(exp(-x^2), x, x)
|
||
```
|
||
|
||
Higher-order derivatives can become involved when the product or quotient rules becomes involved.
|
||
|
||
:::
|
||
|
||
|
||
## Questions
|
||
|
||
|
||
###### Question
|
||
|
||
|
||
The derivative at $c$ is the slope of the tangent line at $x=c$. Answer the following based on @fig-x-expx-sin-pix-over-0-2.
|
||
|
||
|
||
::: {#fig-x-expx-sin-pix-over-0-2}
|
||
```{julia}
|
||
#| echo: false
|
||
fn = x -> -x*exp(x)*sin(pi*x)
|
||
plot(fn, 0, 2)
|
||
```
|
||
|
||
Plot of function $f(x)$ over $[0,2]$
|
||
:::
|
||
At which of these points $c= 1/2, 1, 3/2$ is the derivative negative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["``1/2``", "``1``", "``3/2``"]
|
||
answ = 1
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
Which value looks bigger from reading the graph:
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["``f(1)``", "``f(3/2)``"]
|
||
answ = 2
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
At $0.708 \dots$ and $1.65\dots$ the derivative has a common value. What is it?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
numericq(0, 1e-2)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Consider the graph of the `airyai` function (from `SpecialFunctions`) over $[-5, 5]$ in @fig-plot-airyai-over-minus5-5-identify-sign-of-derivative.
|
||
|
||
::: {#fig-plot-airyai-over-minus5-5-identify-sign-of-derivative}
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
plot(airyai, -5, 5)
|
||
```
|
||
|
||
Plot of `airyai` over $[-5,5]$
|
||
:::
|
||
|
||
At $x = -2.5$ the derivative is positive or negative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["positive", "negative"]
|
||
answ = 1
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
At $x=0$ the derivative is positive or negative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["positive", "negative"]
|
||
answ = 2
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
At $x = 2.5$ the derivative is positive or negative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["positive", "negative"]
|
||
answ = 2
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Compute the derivative of $e^x$ using `limit`. What do you get?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["``e^x``", "``x^e``", "``(e-1)x^e``", "``e x^{(e-1)}``", "something else"]
|
||
answ = 1
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Compute the derivative of $x^e$ using `limit`. What do you get?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["``e^x``", "``x^e``", "``(e-1)x^e``", "``e x^{(e-1)}``", "something else"]
|
||
answ = 4
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Compute the derivative of $e^{e\cdot x}$ using `limit`. What do you get?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = ["``e^x``", "``x^e``", "``(e-1)x^e``", "``e x^{(e-1)}``", "``e \\cdot e^{e\\cdot x}``", "something else"]
|
||
answ = 5
|
||
radioq(choices, answ, keep_order=true)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
In the derivation of the derivative of $\sin(x)$, the following limit is needed:
|
||
|
||
|
||
$$
|
||
L = \lim_{h \rightarrow 0} \frac{\cos(h) - 1}{h}.
|
||
$$
|
||
|
||
This is
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
L" $1$, as this is clearly the analog of the limit of $\sin(h)/h$.",
|
||
L"Does not exist. The answer is $0/0$ which is undefined",
|
||
L" $0$, as this expression is the derivative of cosine at $0$. The answer follows, as cosine clearly has a tangent line with slope $0$ at $x=0$."]
|
||
answ = 3
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Let $f(x) = (e^x + e^{-x})/2$ and $g(x) = (e^x - e^{-x})/2$. Which is true?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
"``f'(x) = g(x)``",
|
||
"``f'(x) = -g(x)``",
|
||
"``f'(x) = f(x)``",
|
||
"``f'(x) = -f(x)``"
|
||
]
|
||
answ= 1
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Let $f(x) = (e^x + e^{-x})/2$ and $g(x) = (e^x - e^{-x})/2$. Which is true?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
"``f''(x) = g(x)``",
|
||
"``f''(x) = -g(x)``",
|
||
"``f''(x) = f(x)``",
|
||
"``f''(x) = -f(x)``"]
|
||
answ= 3
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Consider the function $f$ and its transformation $g(x) = a + f(x)$ (shift up by $a$). Do $f$ and $g$ have the same derivative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
yesnoq("yes")
|
||
```
|
||
|
||
Consider the function $f$ and its transformation $g(x) = f(x - a)$ (shift right by $a$). Do $f$ and $g$ have the same derivative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
yesnoq("no")
|
||
```
|
||
|
||
Consider the function $f$ and its transformation $g(x) = f(x - a)$ (shift right by $a$). Is $g'$ at $x$ equal to $f'$ at $x-a$?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
yesnoq("yes")
|
||
```
|
||
|
||
Consider the function $f$ and its transformation $g(x) = c f(x)$, $c > 1$. Do $f$ and $g$ have the same derivative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
yesnoq("no")
|
||
```
|
||
|
||
Consider the function $f$ and its transformation $g(x) = f(x/c)$, $c > 1$. Do $f$ and $g$ have the same derivative?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
yesnoq("no")
|
||
```
|
||
|
||
Which of the following is true?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
L"If the graphs of $f$ and $g$ are translations up and down, the tangent line at corresponding points is unchanged.",
|
||
L"If the graphs of $f$ and $g$ are rescalings of each other through $g(x)=f(x/c)$, $c > 1$. Then the tangent line for corresponding points is the same.",
|
||
L"If the graphs of $f$ and $g$ are rescalings of each other through $g(x)=cf(x)$, $c > 1$. Then the tangent line for corresponding points is the same."
|
||
]
|
||
answ = 1
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
The rate of change of volume with respect to height is $3h$. The rate of change of height with respect to time is $2t$. At $t=3$ the height is $h=14$ what is the rate of change of volume with respect to time when $t=3$?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
## dv/dt = dv/dh * dh/dt = 3h * 2t
|
||
h = 14; t=3
|
||
val = (3*h) * (2*t)
|
||
numericq(val)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Which equation below is $f(x) = \sin(k\cdot x)$ a solution of ($k > 1$)?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
"``f'(x) = k^2 \\cdot f(x)``",
|
||
"``f'(x) = -k^2 \\cdot f(x)``",
|
||
"``f''(x) = k^2 \\cdot f(x)``",
|
||
"``f''(x) = -k^2 \\cdot f(x)``"]
|
||
answ = 4
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Let $f(x) = e^{k\cdot x}$, $k > 1$. Which equation below is $f(x)$ a solution of?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
"``f'(x) = k^2 \\cdot f(x)``",
|
||
"``f'(x) = -k^2 \\cdot f(x)``",
|
||
"``f''(x) = k^2 \\cdot f(x)``",
|
||
"``f''(x) = -k^2 \\cdot f(x)``"]
|
||
answ = 3
|
||
radioq(choices, answ)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Their are $6$ trig functions. The derivatives of $\sin(x)$ and $\cos(x)$ should be memorized. The others can be derived if not memorized using the quotient rule or chain rule.
|
||
|
||
|
||
What is $[\tan(x)]'$? (Use $\tan(x) = \sin(x)/\cos(x)$.)
|
||
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
trig_choices = [
|
||
"``\\sec^2(x)``",
|
||
"``\\sec(x)\\tan(x)``",
|
||
"``-\\csc^2(x)``",
|
||
"``-\\csc(x)\\cot(x)``"
|
||
]
|
||
radioq(trig_choices, 1)
|
||
```
|
||
|
||
What is $[\cot(x)]'$? (Use $\cot(x) = \cos(x)/\sin(x)$.)
|
||
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
radioq(trig_choices, 3)
|
||
```
|
||
|
||
What is $[\sec(x)]'$? (Use $\sec(x) = 1/\cos(x)$.)
|
||
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
radioq(trig_choices, 2)
|
||
```
|
||
|
||
What is $[\csc(x)]'$? (Use $\csc(x) = 1/\sin(x)$.)
|
||
|
||
|
||
```{julia}
|
||
#| echo: false
|
||
radioq(trig_choices, 4)
|
||
```
|
||
|
||
###### Question
|
||
|
||
|
||
Consider @fig-composition-of-functions-image, a picture of composition.
|
||
|
||
::: {#fig-composition-of-functions-image}
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
f(x) = sin(x)
|
||
g(x) = exp(x)
|
||
a,b = 0, 1.55
|
||
|
||
xs = range(a, b, length=100)
|
||
ys = g.(xs)
|
||
us = range(extrema(ys)..., length=100)
|
||
vs = f.(us)
|
||
pf = plot(vs, us, ylim=extrema(ys), ymirror=true, legend=false)
|
||
xs′ = range(0.5, 1.5, length=100)
|
||
us′ = g.(xs′)
|
||
vs′ = [f(g(1) + g'(1)*(x-1)) for x ∈ xs′]
|
||
plot!(pf, vs′, us′)
|
||
|
||
plot!(pf, [1, f(g(1)), f(g(1))], [g(1), g(1), 1])
|
||
quiver!(pf, [.75, f(g(1))], [g(1), 2], quiver=([-.01, 0],[0, -0.1]))
|
||
pg = plot(xs, ys, ylim=extrema(ys), legend=false)
|
||
plot!(pg, [1,1,0],[1, g(1),g(1)])
|
||
quiver!(pg, [1, 0.5], [2, g(1)], quiver=([0, -0.1],[0.1, 0]))
|
||
plot!(tangent(g,1))
|
||
l = @layout [a b]
|
||
plot(pf, pg, layout=l)
|
||
```
|
||
|
||
Plot of function $f(x)$ rotated so $x$ axis is pointing to the left; plot of $g(x)$ on right. This picture allows graphical composition to be computed.
|
||
:::
|
||
|
||
|
||
The right graph is of $g(x) = \exp(x)$ at $x=1$, the left graph of $f(x) = \sin(x)$ *rotated* $90$ degrees counter-clockwise. Chasing the arrows shows graphically how $f(g(1))$ can be computed. The nearby values $f(g(1+h))$ are – using the tangent line of $g$ at $x-1$ – approximated by $f(g(1) + g'(1)\cdot h)$, as shown in the graph segment on the left.
|
||
|
||
|
||
Assuming the approximation gets better for $h$ close to $0$, as it visually does, the derivative at $1$ for $f(g(x))$ should be given by this limit:
|
||
|
||
|
||
$$
|
||
\begin{align*}
|
||
\frac{d(f\circ g)}{dx}\mid_{x=1}
|
||
&= \lim_{h\rightarrow 0} \frac{f(g(1) + g'(1)h)-f(g(1))}{h}\\
|
||
&= \lim_{h\rightarrow 0} \frac{f(g(1) + g'(1)h)-f(g(1))}{g'(1)h} \cdot g'(1)\\
|
||
&= \lim_{h\rightarrow 0} f'(g(1)) \cdot g'(1).
|
||
\end{align*}
|
||
$$
|
||
|
||
|
||
What limit law, described below assuming all limits exist. allows the last equals sign?
|
||
|
||
|
||
```{julia}
|
||
#| hold: true
|
||
#| echo: false
|
||
choices = [
|
||
raw"""
|
||
The limit of a sum is the sum of the limits:
|
||
``\lim_{x\rightarrow c}(au(x)+bv(x)) = a\lim_{x\rightarrow c}u(x) + b\lim_{x\rightarrow c}v(x)``
|
||
""",
|
||
raw"""
|
||
The limit of a product is the product of the limits:
|
||
``\lim_{x\rightarrow c}(u(x)\cdot v(x)) = \lim_{x\rightarrow c}u(x) \cdot \lim_{x\rightarrow c}v(x)``
|
||
""",
|
||
raw"""
|
||
The limit of a composition (under assumptions on ``v``):
|
||
``\lim_{x \rightarrow c}u(v(x)) = \lim_{w \rightarrow \lim_{x \rightarrow c}v(x)} u(w)``.
|
||
"""
|
||
]
|
||
radioq(choices, 3, keep_order=true)
|
||
```
|