Implicit Differentiation — Mathematics Knowledge System

Overview

Implicit differentiation is a technique for finding dy/dx when y is defined implicitly by an equation relating x and y (e.g. x² + y² = 25), rather than explicitly as y = f(x). The method differentiates both sides of the equation with respect to x, treating y as a function of x and applying the chain rule whenever y appears, then solves algebraically for dy/dx.

Intuition

Many curves — circles, ellipses, and more complex shapes — cannot be written as a single explicit function y=f(x), yet they still have well-defined tangent lines at most points. Implicit differentiation lets you find the slope of the tangent without first solving for y, by remembering that y is secretly a function of x and every time you differentiate a term involving y, the chain rule tacks on a dy/dx factor.

Formal Definition

Definition

Given an equation F(x,y) = G(x,y), differentiate both sides with respect to x, treating y as y(x):

\frac{d}{dx}\big[y^n\big] = n y^{n-1}\frac{dy}{dx}

Chain rule applied to any power of y

\frac{d}{dx}\big[f(x)g(y)\big] = f'(x)g(y) + f(x)g'(y)\frac{dy}{dx}

Product rule combined with the chain rule when a term mixes x and y

Applications

Implicit differentiation finds tangent lines and rates of change for orbit equations and other physical curves not expressible as y=f(x).

Worked Examples

Differentiate both sides with respect to x, applying the chain rule to y².
$2x + 2y\frac{dy}{dx} = 0$
Solve for dy/dx.
$\frac{dy}{dx} = -\frac{x}{y}$

Answer: dy/dx = -x/y

Practice Problems

Difficulty 4/10

Find dy/dx if x²y + y³ = 10.

Difficulty 5/10

Find the slope of the tangent line to x² + xy + y² = 7 at the point (1,2).

Common Mistakes

Common Mistake

Forgetting to multiply by dy/dx when differentiating a term containing y.

Every time you differentiate an expression involving y with respect to x, the chain rule requires an extra factor of dy/dx — e.g. d/dx[y²] = 2y(dy/dx), not just 2y.

Common Mistake

Forgetting the product rule on mixed terms like xy.

d/dx[xy] = y + x(dy/dx) (product rule), not simply dy/dx or y alone.

Summary

Implicit differentiation finds dy/dx when y is defined implicitly by an equation in x and y, without solving for y explicitly.
Differentiate both sides of the equation with respect to x, treating y as y(x) and applying the chain rule to every y-term.
Terms mixing x and y (like xy) require the product rule combined with the chain rule.
After differentiating, collect all dy/dx terms on one side and solve algebraically.

References

WebsiteWikipedia — Implicit function