Implicit Differentiation Calculator
Implicit differentiation calculator. Solve dy/dx from an equation F(x, y) = 0, including circles and other relations that are not functions.
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<iframe src="https://derivativecalculatorai.com/implicit-differentiation-calculator" title="Derivative Calculator AI" width="100%" height="640" style="border:0"></iframe>Some curves are not functions y = f(x). The circle x^2 + y^2 = 25 has two y values for most x. Implicit differentiation still produces the slope: rewrite as F = x^2 + y^2 − 25, then dy/dx = −(∂F/∂x) / (∂F/∂y) = −x/y.
The method is the chain rule in disguise. Every y is a function of x, so differentiating y^2 contributes 2 y dy/dx. You then solve that equation for dy/dx. The partials are only a bookkeeping device that isolates the coefficient of dy/dx.
Enter one equation containing both x and y. The calculator returns dy/dx. If the partial with respect to y is 0, the relation does not determine a slope.
Rule and examples
Frequently asked questions
What is implicit differentiation?
It finds dy/dx from an equation that mixes x and y, without solving for y first.
What is dy/dx for x^2 + y^2 = 25?
−x/y, wherever y is not zero. The top and bottom of the circle are horizontal; the left and right are vertical.
Why do partial derivatives appear?
dy/dx = −Fx/Fy. Fx and Fy are partials of F = left − right. They are not a second, unrelated topic.
What if the equation has no y?
Then it does not define y as a function of x. Use the ordinary derivative calculator instead.