2.5: The Chain Rule

We further build our understanding of the derivative by looking at compositions of functions next.

For example, how would you take the derivative of $F(x) = \sin(x^2)$?

Chain Rule
If $h(x) = f(g(x))$, then \[h'(x) = \dfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)\] If we write $u = g(x)$, meaning $y = h(x) = f(g(x)) = f(u)$, then \[\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot \dfrac{du}{dx}\]

In other words, $h'(x)$ is the derivative of the outside function evaluated at the inside function, then multiplied by the derivative of inside.

Let $F(x) = (3x+1)^2$.
  1. Find $F'(x)$ using the chain rule.
  2. Find $F'(x)$ without using the chain rule.
Differentiate the following:
  1. $F(x) = \sqrt{x^2 + 1}$
  2. $F(x) = \sin(x^2)$
  3. $F(x) = \sin^2(x)$

The chain rule is used when a function is taken to a power, for example \[(2x+5)^5 \qquad \sqrt{4x^2 + 2x + 1} \qquad (4x^5 + 3x^2 + 1)^{6/5}\] This means the outside function $f(x) = x^n$ sometimes! Thus:

The General Power Rule
If $g$ is differentiable and $h(x) = [g(x)]^n$, then \[h'(x) = \dfrac{d}{dx}[g(x)]^n = n[g(x)]^{n-1}g'(x)\]
Differentiate $y = (x^3-1)^{100}$.
Differentiate the following:
  1. $F(x) = \dfrac{1}{\sqrt[3]{x^2 + x + 1}}$
  2. $H(t) = \left(\dfrac{t - 2}{2t + 1}\right)^9$

Longer Chains

Suppose $y = f(g(h(t)))$. Assigning $y = f(u), u = g(x), x = h(t)$, what is $\dfrac{dy}{dt}$?

Differentiate the following:
  1. $F(x) = \sin(\cos(\tan(x)))$