AP®MathsCollege BoardCalculus ABStudy GuidesUnit 3: Differentiation of Composite, Implicit & Inverse FunctionsUnit 3 OverviewUnit 3 Summary

Unit 3 Summary (College Board AP® Calculus AB): Study Guide

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 5 May 2026

Differentiation of composite, implicit & inverse functions summary

Key definitions

$f^{(n)} (x)$ and $\frac{d^{n}}{d x^{n}} (f (x))$ denote the nth derivative, which is the result when the function is differentiated $n$ times
An equation is given implicitly if it is not written in the form $y = f (x)$

Key theorems

The chain rule (for differentiating composite functions)
- If $h (x) = f (g (x))$
- Then $h' (x) = f' (g (x)) \cdot g' (x)$
The inverse function theorem (for differentiating inverse functions)
- If $g (x) = f^{- 1} (x)$
- Then $g' (x) = \frac{1}{f^{'} (g (x))}$
- This can be written as $\frac{d x}{d y} = \frac{1}{\frac{d y}{d x}}$
The chain rule is used to differentiate terms in an implicit equation
- $\frac{d}{d x} (f (y)) = f' (y) \cdot \frac{d y}{d x}$

Key formulas

Derivatives of inverse functions

$f (x)$	$f' (x)$
$\sin^{- 1} x$	$\frac{1}{\sqrt{1 - x^{2}}}$ , $- 1 < x < 1$
$\cos^{- 1} x$	$- \frac{1}{\sqrt{1 - x^{2}}}$ , $- 1 < x < 1$
$\tan^{- 1} x$	$\frac{1}{1 + x^{2}}$
$\csc^{- 1} x$	$- \frac{1}{\|x\| \sqrt{x^{2} - 1}}$ , $x < - 1$ or $x > 1$
$\sec^{- 1} x$	$\frac{1}{\|x\| \sqrt{x^{2} - 1}}$ , $x < - 1$ or $x > 1$
$\cot^{- 1} x$	$- \frac{1}{1 + x^{2}}$

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