Implicit Differentiation Calculator

Enter an implicit equation (e.g. x³ + y³ = 2xy) and choose whether to differentiate with respect to x or y. The Implicit Differentiation Calculator computes dy/dx (or dx/dy), applies the chain rule step by step, and optionally finds the second derivative. You can also evaluate the derivative at a specific point (x₀, y₀) to get a numeric slope value. Also try the Horizontal Tangent Line Calculator.

Enter the left-hand side of your implicit equation. Use ^ for powers, * for multiplication.

Enter the right-hand side of your implicit equation.

Differentiate with Respect to *

Also find the second derivative? *

Leave blank to skip point evaluation.

Both x₀ and y₀ are needed for numeric evaluation.

Results

dy/dx (Derivative Expression)

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Slope at (x₀, y₀)

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Second Derivative Note

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∂F/∂x at Point

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∂F/∂y at Point

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Results Table

What is implicit differentiation?

Implicit differentiation is a technique for finding the derivative of a function that is not expressed explicitly as y = f(x), but instead defined by an equation relating x and y (e.g. x² + y² = 25). You differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule wherever y appears, then solve for dy/dx. See also our Arc Length Calculator (Calculus).

When should I use implicit differentiation?

Use implicit differentiation when it is difficult or impossible to isolate y explicitly. Common situations include circles, ellipses, and curves like x³ + y³ = 6xy. It is also used in related-rates problems in physics and engineering where two quantities change with respect to a third variable such as time.

How does the calculator find dy/dx for an implicit equation?

The calculator differentiates both sides of your equation with respect to x, treating y as an implicit function. It applies the chain rule to every term containing y (contributing a dy/dx factor), collects all dy/dx terms on one side, and solves algebraically. The result is expressed as a ratio of partial derivatives: dy/dx = −Fx/Fy.

What does 'evaluate at a point (x₀, y₀)' mean?

After finding the symbolic expression for dy/dx, you can substitute specific numeric values for x and y to get the slope of the tangent line at that particular point on the curve. Enter x₀ and y₀ in the optional fields, and the calculator will output a single numeric slope value. You might also find our calculate Critical Points Found, Local Maximum — x value & Local Maximum — f(x) value — Maxima and Minima useful.

How is the second derivative d²y/dx² calculated implicitly?

Finding d²y/dx² implicitly requires differentiating dy/dx again with respect to x, treating both y and dy/dx as functions of x. The result typically still contains dy/dx, which is then substituted with the first-derivative expression. Enable the second derivative toggle to see this computed automatically.

Can I differentiate with respect to y instead of x?

Yes. If you need dx/dy instead of dy/dx — for example, when studying inverse relationships — simply select 'y' as the differentiation variable. The calculator will then treat x as an implicit function of y and apply the chain rule accordingly.

What common mistakes should I avoid with implicit differentiation?

The most common mistake is forgetting to apply the chain rule when differentiating terms with y — every derivative of y with respect to x must include a dy/dx factor. Other pitfalls include sign errors when moving terms across the equals sign and forgetting to use the product rule on terms like xy.

How do I enter expressions correctly in this calculator?

Use standard algebraic notation: ^ for exponentiation (e.g. x^3), * for multiplication (e.g. 2*x*y), and parentheses to group terms (e.g. (x+y)^2). Trigonometric functions are written as sin(x), cos(y), tan(x), etc. Separate the two sides of your equation into the left and right input fields — do not include the equals sign.