Implicit Differentiation

Unit 3 · Differentiation — Composite, Implicit, Inverse

Implicit Differentiation

When a curve is defined by an equation involving both x and y, differentiate both sides with respect to x, treating y as a function of x. Every time you differentiate y, multiply by dy/dx (Chain Rule).

Key Principle

Differentiate both sides w.r.t. x; for any y term:

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

AP Tip: After differentiating, collect all dy/dx terms on one side, then factor out dy/dx and divide.

Type 1

Finding dy/dx Implicitly

Differentiate both sides with respect to x, apply product rule where needed, collect dy/dx terms, and solve.

Example 1

Find dy/dx for the curve.

x^2 + 3xy + y^2 = 7

Example 2

Find dy/dx.

x^2 y + y^3 = 5

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Type 2

Tangent Line to an Implicit Curve

Find dy/dx implicitly, evaluate it at the given point to get the slope, then use point-slope form.

Example 3

Find the tangent line to the circle at (3, 4).

x^2 + y^2 = 25

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Type 3

Second Derivative Implicitly

Differentiate dy/dx again with respect to x using the quotient rule, then substitute the expression for dy/dx to write d²y/dx² in terms of x and y only.

Example 4

Find d²y/dx² for the circle. Express in terms of x and y.

x^2 + y^2 = 16

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