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Implicit Differentiation Calculator

Implicit derivative calculator. Enter F(x,y)=0 and a point to compute the tangent slope dy/dx=−Fx/Fy, with the tangent line—for implicit differentiation.

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F(x,y)=0 ⇒ dy/dx = − Fx / Fy
Use x and y; e.g. circle x^2+y^2-25
Point on the curve
Point on the curve

📖 Tutorial｜Implicit Differentiation

1. Definition
For an implicit F(x,y)=0, differentiate and solve for y′: dy/dx = −Fx/Fy (partial derivatives).
2. Symbols
SymbolMeaning
F(x,y)Left side of the equation
Fx, FyPartial derivatives
(x0,y0)A point on the curve
dy/dxTangent slope
3. How it works
  • We numerically compute Fx and Fy;
  • Apply dy/dx=−Fx/Fy;
  • When Fy=0 the tangent is vertical.
4. Steps
  1. Enter F(x,y), e.g. x^2+y^2-25;
  2. Enter a point (x0,y0) on the curve;
  3. Click Calculate for the tangent slope.
5. Example
Example: circle x²+y²=25 at (3,4). Fx=6, Fy=8, dy/dx=−6/8=−0.75.
6. Pitfalls
Use a point actually on the curve (F=0);
Fy=0 means a vertical tangent, dy/dx undefined;
Do not forget the minus sign.

❓ FAQ｜Implicit Differentiation

Why the minus sign?
Differentiate F(x,y(x))=0: Fx + Fy·y′=0, so y′=−Fx/Fy; the sign comes from rearranging.
What must the point satisfy?
F(x0,y0)=0; it must actually lie on the implicit curve.
What does Fy=0 mean?
The tangent is vertical, so dy/dx is infinite.
Implicit vs explicit differentiation?
Explicit y=f(x) is differentiated directly; implicit differentiates the whole equation and solves for y′.
Which implicit equations are supported?
Any F(x,y)=0 written in x and y: circles, ellipses, hyperbolas, y=cos(xy), etc.