The Chain Rule The Chain Rule
Introduction
We will prove that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)
Here is an example:
f(x)=\sin(x) \quad g(x)=x^2 \quad f(g(x))=\sin(x^2)
f'(g(x))=\cos(x^2) \quad g'(x)=2x
\frac{d}{dx}\sin(x^2)=\cos(x^2)2x
The derivative of a function is the change in the function in relation to the change in the input.
For example,
\frac{d}{dx}f(x)
is the change in
f(x)
as
x
changes.
Proving
\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)
is proving the change in
f(g(x))
as
x
changes is the same as the change in
f(g(x))
as
g(x)
changes mulitplied with the change in
g(x)
as
x
changes.
In notation, this is:
\frac{d}{dx}f(g(x))=\frac{d}{dg(x)}f(g(x)) \cdot \frac{d}{dx}g(x)
Proof of the Chain Rule
\begin{align*}
\frac{d}{dx}f(x)&=\lim_{x \to a}\frac{f(x)-f(a)}{x-a} &\quad\quad \frac{d}{dx}g(x)&=\lim_{x \to a}\frac{g(x)-g(a)}{x-a} \\\\
\frac{d}{dx}f(g(x))&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{x-a} &\quad\quad \frac{d}{dg(x)}f(g(x))&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{g(x)-g(a)} \\\\\\
\end{align*} \\
\begin{align*}
\frac{d}{dg(x)}f(g(x)) \cdot \frac{d}{dx}g(x)&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{g(x)-g(a)} \cdot \lim_{x \to a}\frac{g(x)-g(a)}{x-a} \\\\
&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{g(x)-g(a)} \cdot \frac{g(x)-g(a)}{x-a} \\\\
&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{x-a} \\\\
&=\frac{d}{dx}f(g(x))
\end{align*}
Therefore, the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied with the derivative of the inner function.
In other words, finding the rate that
f
changes as
g
changes and multiplying it with the rate
g
changes as
x
changes is the same as finding the rate
f
changes as
x
changes.
Examples
Example 1: Find the derivative of
y = (2x + 1)^5
.
\begin{align*}
y &= (2x + 1)^5 \\\\
\frac{dy}{dx} &= 5(2x + 1)^4 \, \frac{d}{dx}(2x + 1) \\\\
&= 5(2x + 1)^4 \cdot 2 \\\\
&= 10(2x + 1)^4
\end{align*}
Example 2: Find the derivative of
f(x) = \sin(x^2)
.
\begin{align*}
f(x) &= \sin(x^2) \\\\
f'(x) &= \cos(x^2) \cdot \frac{d}{dx}(x^2) \\\\
&= \cos(x^2) \cdot 2x \\\\
&= 2x\cos(x^2)
\end{align*}
Example 3: Find the derivative of
g(t) = \sqrt{t^3 + 1}
.
\begin{align*}
g(t) &= \sqrt{t^3 + 1} = (t^3 + 1)^{\frac{1}{2}} \\\\
g'(t) &= \frac{1}{2}(t^3 + 1)^{-\frac{1}{2}} \cdot \frac{d}{dt}(t^3 + 1) \\\\
&= \frac{1}{2}(t^3 + 1)^{-\frac{1}{2}} \cdot 3t^2 \\\\
&= \frac{3t^2}{2\sqrt{t^3 + 1}}
\end{align*}
Example 4: Find the derivative of
h(x) = e^{\tan(x)}
.
\begin{align*}
h(x) &= e^{\tan(x)} \\\\
h'(x) &= e^{\tan(x)} \cdot \frac{d}{dx}(\tan(x)) \\\\
&= e^{\tan(x)} \cdot \sec^2(x) \\\\
&= \sec^2(x)e^{\tan(x)}
\end{align*}