Chain Rule

Composition of Functions

We define a composite (or compound) function of $x$ as a function that is "composed" of two functions;  one function depends on the other function that depends on $x$, namely $y= f(g(x))$.   The notation you may have seen before is $y=(f \circ g)(x)=f(g(x))$.  Suppose that we have variables $x$, $y$, and $u$, and that $y = f(u)$ and $u = g(x)$.  (Previously we called $u=g(x)$ the inside part of $f(g(x))$.)  Then $f$ is a composite function of $x$.

DO:  Find $f$ and $g$ and write as $f(g(x))$ in the following examples in order to familiarize yourself with the notation.  Determine the inside part and outside part of each.

1. $y=\sin(x^2)$ is a composite function with $u=x^2$ and $y=\sin(u)$.


2. $y=\sin^2(x)$ is a composite function with $u=\sin(x)$ and $y=u^2$.


3. $y=e^{3x}$ is a composite function with $u=3x$ and $y=e^u$.


4. $y=(x^2 + 4x + 7)^5$ is a composite function with $u=x^2 + 4x + 7$ and $y=u^5$.

Note that $\displaystyle \frac{dy}{dx}$ is the same as $\displaystyle{\frac{d}{dx}\Big(f\big(g(x)\big)\Big)}$, that $\displaystyle{\frac{dy}{du}} = f'(u)=f'(g(x))$, and that $\displaystyle{\frac{du}{dx}}$ is the same thing as $g'(x)$. So Version 2 says the exact same thing as Version 1.

Let's see how it applies to our examples above.

DO:  After looking at the first example, try to do others before looking at the solutions.  You will learn much more by trying and doing than by reading.  While you read, write down what you are thinking and what is happening.