AP Calculus AB and BC glossary

Inverse function derivative

The derivative of an inverse function at a point equals the reciprocal of the original function's derivative, evaluated at the matching point. You never need a formula for the inverse itself, only the point correspondence.

(f−1)′(b)=1f′(f−1(b))\left(f^{-1}\right)'(b) = \frac{1}{f'\left(f^{-1}(b)\right)}

The geometry is the reason: reflecting a graph across the line y=xy = x swaps rise and run, so slopes invert. If ff passes through (2,7)(2, 7) with slope 4, then f−1f^{-1} passes through (7,2)(7, 2) with slope 14\frac{1}{4}.

The reliable procedure is to find the input aa with f(a)=bf(a) = b, compute f′(a)f'(a), and take its reciprocal. Almost all the difficulty in these problems is bookkeeping about which point goes where.

The mistake

Evaluating f′f' at bb instead of at f−1(b)f^{-1}(b). The inner evaluation point belongs to the original function's domain, not the inverse's.

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