Implicit differentiation finds $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation, without first solving for $y$ explicitly. It's especially useful when solving for $y$ is hard or impossible.

Procedure: differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (so each $y$ term gets a $\frac{dy}{dx}$ via chain rule), then solve for $\frac{dy}{dx}$ .

Example: For $x^2 + y^2 = 25$ (a circle):

Differentiate both sides: $2x + 2y \frac{dy}{dx} = 0$ .
Solve: $\frac{dy}{dx} = -\frac{x}{y}$ .

This gives the slope at any point on the circle without needing $y = \pm\sqrt{25 - x^2}$ .

Implicit differentiation is the standard tool for:

Tangent lines to curves that are not graphs of functions.
Related rates problems (water filling a cone, ladder sliding down a wall).
Differentiating inverse functions (the derivation of $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$ uses it).
Solving differential equations and curves of constant property (level curves).

Implicit Differentiation

Implicit differentiation finds dy/dx when y is defined implicitly by an equation (like x²+y²=25), without first solving for y.