Derivative of a Composite Function: The Chain Rule
Big Idea
When functions are nested inside each other — like $\sin(x^2)$ or $(3x+1)^5$ — we need a special technique called the Chain Rule to find their derivatives. It's one of the most powerful and frequently used rules in all of calculus.
What is a Composite Function?
A composite function is formed when one function is applied to the result of another function. If we have two functions $f$ and $g$, the composite function is written as:
$$(f \circ g)(x) = f(g(x))$$
Here, $g(x)$ is the inner function and $f(u)$ is the outer function, where $u = g(x)$.
| Composite Function $f(g(x))$ | Outer Function $f(u)$ | Inner Function $g(x)$ |
|---|---|---|
| $\sin(x^2)$ | $\sin(u)$ | $u = x^2$ |
| $(3x + 1)^5$ | $u^5$ | $u = 3x + 1$ |
| $e^{2x+3}$ | $e^u$ | $u = 2x + 3$ |
| $\ln(\cos x)$ | $\ln(u)$ | $u = \cos x$ |