3.3: The Chain Rule

Chain Rule

How about taking the derivative of a composition of functions?

Consider $y = h(x) = g(f(x))$. Suppose we let a new variable $u$ exist and $u = f(x)$.

Then $y = g(f(x)) = g(u)$. Picture:

We can compute $g'(u)$; it is $\dfrac{dy}{du}$.

We can also compute $f'(x)$; it is $\dfrac{du}{dx}$.

How do we put these parts together?

Chain Rule
If $h(x) = g(f(x))$, then \[h'(x) = \dfrac{d}{dx}g(f(x)) = g'(f(x))\cdot f'(x)\] If we write $u = f(x)$, meaning $y = h(x) = g(f(x)) = g(u)$, 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. $F(x) = x^2(2x+3)^5$
  3. $F(x) = (2x^2 + 3)^4(3x-1)^5$
  4. $F(x) = \dfrac{1}{(4x^2 - 7)^2}$
The membership of your local gym is approximated by the function \[N(t) = 100(64 + 4t)^{2/3} \qquad 0 \leq t \leq 52\] where $N(t)$ gives the number of members at the beginning of week $t$.
  1. Find $N'(t)$.
  2. How fast was the gym's membership increasing initially ($t = 0$)?