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Implicit differentiation

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

Implicit differentiation

A technique for finding the derivative of an implicit equation (ex: $y = x y^{2} + 1$ ) that treats $y$ as some function of $x$ , so that $\frac{d}{d x} f (y) = f^{'} (y) \cdot \frac{d y}{d x}$

Example 1: Differentiate $y = x + y^{2} + 1$

$\frac{d}{d x} (y = x + y^{2} + 1) \Rightarrow \frac{d y}{d x} = 1 + 2 y \frac{d y}{d x} + 0$

$\frac{d y}{d x} (1 - 2 y) = 1 \Rightarrow \frac{d y}{d x} = \frac{1}{1 - 2 y}$

Related Glossary Terms

Chain rule, Derivative

Index

Chapter 4 - FFR - Implicit Differentiation

Chapter 5 - Powers and Logarithms

Chapter 5 - Inverse Functions

Chapter 6 - Optimization

Chapter 6 - Related Rates

Glossary and deeper reading · 1 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Implicit differentiation
Implicit differentiation finds a derivative from an equation relating variables without first solving explicitly for one variable. When y depends on x, differentiating a term containing y requires the chain rule and a factor of dy/dx. For example, x²+y²=1 gives 2x+2y(dy/dx)=0. Solving that relation for dy/dx gives −x/y only where y≠0 and the relation supports the local function being considered.
Read more: 3.2 · Implicit differentiation →

Implicit differentiation

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