What this implicit differentiation calculator does
Explicit derivatives assume y = f(x). Many homework curves are given implicitly —
circles, ellipses, folium curves, or products mixed with exponentials — as
F(x, y) = 0. An implicit derivative calculator helps you
find dy/dx without solving for y first.
This page uses the same differentiation engine as our partial derivative calculator. You compute the two building blocks Fₓ and Fᵧ with labeled steps, then combine them by hand into the classic formula. That split matches how textbooks present the method and keeps every rule visible.
How to find dy/dx with this tool
-
Rewrite the equation so one side is zero. Example: x² + y² = 1 becomes
x^2 + y^2 - 1. -
Set the variable to
xand Calculate — that run treats y as constant and returns Fₓ (partial with respect to x). -
Change the variable to
yand Calculate again for Fᵧ. - Form dy/dx = −Fₓ / Fᵧ. Where Fᵧ = 0, check vertical tangents or singular points separately.
Prefer explicit single-variable work? Use the main derivative calculator. Prefer seeing every product and chain label on a solved-for-y function? Open the derivative calculator with steps.
Worked implicit differentiation examples
Circle: x² + y² = 1
F = x^2 + y^2 - 1. Then Fₓ = 2x and Fᵧ = 2y, so dy/dx = −x/y for y ≠ 0.
Run both partials above and match the steps to your notes.
Product curve: x²y + y³ = 4
F = x^2*y + y^3 - 4. Fₓ = 2xy and Fᵧ = x² + 3y², giving
dy/dx = −2xy / (x² + 3y²). The quotient is assembled after the two calculator runs.
Transcendental mix: xy + e^y = 0
Use x*y + e^y (or move a constant if needed). Fₓ = y and
Fᵧ = x + e^y, so dy/dx = −y / (x + e^y). The exponential’s chain-style factor appears
when you differentiate with respect to y.
Why textbooks use Fₓ and Fᵧ
Differentiating F(x, y(x)) = 0 with the multivariable chain rule yields Fₓ + Fᵧ · y′ = 0, which rearranges to y′ = −Fₓ/Fᵧ. Computing the partials separately avoids expanding every y′ term by hand on messy products — and this free implicit differentiation calculator shows those partial steps so you can still audit the algebra.
Second derivatives of implicit curves need another differentiation of y′ (quotient or product rule). Start with the first derivative here, then use the second derivative calculator on an explicit form of y′ if your course asks for d²y/dx².
Tips for accurate input
Always bring constants to the left so the expression represents F, not “left = right.”
Use * for products like x*y, and wrap sums inside powers or
denominators. If your relation is already solved for y, you do not need implicit mode —
differentiate y directly on the homepage instead.
Implicit derivative calculator FAQ
What is implicit differentiation?
When a relation is given as F(x, y) = 0 instead of y = f(x), you differentiate both sides (or use partials) and solve for dy/dx without isolating y.
How do I use this implicit derivative calculator?
Enter F, compute ∂F/∂x and ∂F/∂y by switching the variable, then form dy/dx = −Fₓ/Fᵧ. The labeled steps document each partial.
Is this the same as an implicit differentiation calculator?
Yes. People also search “derivative implicit calculator” and “calculate the derivative using implicit differentiation” for the same homework task.
Is this tool free?
Yes. Partials and step-by-step explanations are free — no account, no Pro wall.
When should I use partials instead of d/dx with y′?
Both are valid. The Fₓ/Fᵧ route is compact for complicated F. Expanding with y′ by hand is fine for short products. Use whichever your instructor grades.