Math · Calculus I · Concept

Implicit differentiation

Implicit differentiation finds dy/dx when y is tied to x by an equation you have not solved for y. Differentiate both sides with respect to x, multiply the derivative of every y term by dy/dx (the chain rule), then solve for dy/dx.

When y is not isolated

An equation such as x² + y² = 25 describes a curve without giving y as one formula in x. Near most of its points, though, the curve is the graph of some differentiable function y(x), so dy/dx still means the slope of the tangent there.

x^{2} + y^{2} = 25

Differentiate both sides with respect to x

Treat y as a function of x. A term in x differentiates as usual. A term in y needs the chain rule: the derivative of y² is 2y·dy/dx, because y itself changes with x. Both sides stay equal as x changes, so their derivatives are equal.

\frac{d}{d x} (y^{2}) = 2 y \frac{d y}{d x}

Solve for dy/dx

Collect the terms that contain dy/dx on one side, factor dy/dx out and divide. The result usually involves both x and y, which is expected: the slope depends on which point of the curve you are at.

2 x + 2 y \frac{d y}{d x} = 0 ⟹ \frac{d y}{d x} = - \frac{x}{y}

\begin{array}{c} 2 x + 2 y \frac{d y}{d x} = 0 \\ ⟹ \frac{d y}{d x} = - \frac{x}{y} \end{array}

Products need the product rule

A term such as xy is a product of two functions of x, so its derivative is 1·y + x·dy/dx. Leaving out the second term is the most common error in implicit differentiation.

\frac{d}{d x} (x y) = y + x \frac{d y}{d x}

Evaluate at a point on the curve

Substitute the coordinates of a point that lies on the curve. Where the formula has a zero denominator, as at (5, 0) on the circle, the tangent is vertical and dy/dx is undefined.

Common mistakes

Differentiating y² as 2y, without the factor dy/dx.
Differentiating xy as y alone, or as dy/dx alone, instead of y + x·dy/dx.
Substituting a point that is not on the curve.
Giving the constant on the right a nonzero derivative: the derivative of 25 is 0.

Key terms

Implicit differentiation: Finding dy/dx from an equation in x and y without solving for y: differentiate both sides with respect to x, multiply the derivative of each y term by dy/dx, then solve for dy/dx.
Implicit relation: An equation in x and y, such as x² + y² = 25, that isn’t solved for y. It can describe curves that no single function y = f(x) covers.
Chain rule: The rule for differentiating a function inside another function: differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside.
Tangent line: The line that matches a curve’s direction at a point, with slope equal to the derivative there. It can cross the curve; in trigonometry, “tangent” also names tan θ = sin θ/cos θ.
Derivative: The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.

Work through an example

For the circle x² + y² = 25, find dy/dx by implicit differentiation, then the tangent line at (3, 4).

Find dy/dx on a circle and its tangent line →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Calculus Volume 1 — Implicit differentiation

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Plot the circle in Implicit curves Graph the circle and its tangent Write the steps in Math Open worked example on a board Implicit differentiation in Math Reference

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