∂Implicit Differentiation Calculator
Enter an equation in x and y to find dy/dx, with an optional point for the numeric slope.
Use ^ for powers and juxtaposition or * for multiplication (e.g. 3x^2y or x^2*y^3). Constants like π aren't supported.
Free Implicit Differentiation Calculator
This implicit differentiation calculator handles polynomial equations in x and y — like x² + y² = 25 — where y isn't isolated as a function of x. It moves everything to one side to form F(x, y) = 0, finds the partial derivatives ∂F/∂x and ∂F/∂y term by term, and applies the implicit function rule: dy/dx = −(∂F/∂x) / (∂F/∂y).
Enter a point (x₀, y₀) on the curve to get the numeric slope at that location — handy for tangent line problems on circles, ellipses, and other implicit curves that can't be solved for y directly.
Free, runs entirely in your browser, no sign-up required.
About the Implicit Differentiation Calculator
Features
Mixed x/y terms
Handles terms like x²y³ using the product rule automatically.
Numeric slope
Enter a point to see dy/dx evaluated there.
Step-by-step
See F(x,y), both partial derivatives, and the final formula.
Instant, private
Runs entirely in your browser. No data is sent anywhere.
Frequently Asked Questions
What is implicit differentiation?
Implicit differentiation finds dy/dx for an equation relating x and y that isn't solved for y as an explicit function of x, such as x² + y² = 25.
How does the calculator find dy/dx?
It rewrites the equation as F(x, y) = 0, computes the partial derivatives ∂F/∂x and ∂F/∂y term by term, then applies dy/dx = −(∂F/∂x) / (∂F/∂y).
Why is there a negative sign in the formula?
It comes from the implicit function theorem — differentiating F(x, y) = 0 with respect to x using the chain rule gives ∂F/∂x + (∂F/∂y)(dy/dx) = 0, which rearranges to dy/dx = −(∂F/∂x) / (∂F/∂y).
What input format does the equation need?
Standard polynomial terms in x and y with ^ for exponents, an = sign, and + or − between terms, such as x^2 + y^2 = 25 or 3x^2*y - y^3 = 4.
What does it mean when the slope is undefined?
If ∂F/∂y = 0 at your point, the tangent line is vertical, and dy/dx is undefined at that exact location — this happens, for example, at the leftmost and rightmost points of a circle.
Do I need to enter a point?
No — the point (x₀, y₀) is optional. Without it, you still get the general dy/dx formula; with it, you also get the numeric slope at that specific point.
Does the point need to satisfy the equation?
For the slope to be meaningful for the curve, yes — the point should lie on the curve. The calculator evaluates the derivative at whatever (x₀, y₀) you enter regardless, so double-check the point satisfies your equation.
Can this handle terms with both x and y, like x²y³?
Yes — mixed terms are differentiated using the product rule automatically, since the tool tracks the power of x and y in every term separately.
Why is this used instead of solving for y first?
Many curves — circles, ellipses, and more complex implicit curves — can't be algebraically solved for y as a single explicit function of x, so implicit differentiation is the only practical way to find the slope.
Is this calculator free to use?
Yes, completely free with no sign-up, and it runs entirely in your browser.