AP Calculus AB and BC glossary

Inner function

In a composition, the inner function is the one applied first, sitting inside the parentheses. The chain rule differentiates the outer function while holding the inner one fixed, then multiplies by the derivative of the inner function.

\frac{d}{dx}f\left(g(x)\right) = f'\left(g(x)\right)\cdot g'(x)

In $\sin\left(x^{2}\right)$ the inner function is $x^{2}$ and the outer is $\sin$ . Naming them before differentiating is the single most reliable way to avoid dropping the chain rule factor.

The mistake

Forgetting the extra factor entirely. The derivative of sine of x squared is cosine of x squared times two x, not just cosine of x squared.

Appears in: Unit 3: Chain Rule, Implicit, and Inverses