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Implicit Differentiation Calculator

dy/dx for equations like x² + y² = 25 — via −Fx/Fy, steps shown.

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How Implicit Differentiation Works

Some curves cannot be written as y = (something with only x) — think of a circle x² + y² = 25.Implicit differentiation differentiates both sides of the equation as written, remembering that y is secretly a function of x (so differentiating y² gives 2y·dy/dx by the chain rule), then solves for dy/dx.

The Shortcut Formula

Write the equation as F(x, y) = 0. Then dy/dx = −Fx/Fy, where Fx and Fy are the ordinary partial derivatives. For the circle: Fx = 2x, Fy = 2y, so dy/dx = −x/y — exactly the slope of the tangent at any point on the circle.

When It Fails

The formula needs Fy ≠ 0. At points where Fy = 0 the tangent is vertical and dy/dx is undefined. For ordinary single-variable derivatives, see our derivative calculator.

Frequently Asked Questions

What is implicit differentiation?

It differentiates both sides of an equation like x² + y² = 25 without first solving for y, using the chain rule on y-terms, then solves the result for dy/dx.

What is the formula for dy/dx?

Write the equation as F(x,y) = 0. Then dy/dx = −Fx/Fy, where Fx and Fy are the partial derivatives of F. For the circle this gives dy/dx = −x/y.

When does implicit differentiation fail?

When Fy = 0 at the point of interest — the tangent is vertical there, so dy/dx is undefined.

Is this implicit differentiation calculator free?

Yes — every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.