Implicit Differentiation Calculator

How to Use This Calculator

1. Enter F(x,y) set equal to zero (e.g. x^2+y^2-25 for the unit circle).
2. Enter the point (x₀, y₀).
3. Get dy/dx, the tangent line equation, and partial derivatives Fₓ, F_y.
4. Use Common Curves tab for circle, ellipse, and hyperbola presets.
5. The Professional tab adds d²y/dx² and normal line.

Formula

Example

Frequently Asked Questions

• What is implicit differentiation? ▼
  Implicit differentiation finds dy/dx when y is defined implicitly by F(x,y)=0. Differentiate both sides with respect to x (treating y as a function of x), then solve for dy/dx.
• What is the formula for implicit differentiation? ▼
  By the implicit function theorem: dy/dx = −(∂F/∂x)/(∂F/∂y) = −Fₓ/F_y, provided F_y ≠ 0. This is the quickest formula when F(x,y) is given explicitly.
• How do I find the tangent line to a circle? ▼
  For x²+y²=r², dy/dx = −x/y. At point (x₀,y₀): tangent line is y−y₀ = (−x₀/y₀)(x−x₀), or equivalently x₀x + y₀y = r².
• When does a vertical tangent occur? ▼
  A vertical tangent occurs when F_y = ∂F/∂y = 0 at the point (but F_x ≠ 0). The slope dy/dx is undefined there.
• How do I find the second derivative d²y/dx² implicitly? ▼
  d²y/dx² = −(Fₓₓ·F_y² − 2Fₓᵧ·Fₓ·F_y + F_yy·Fₓ²)/F_y³. Our Professional tab computes this numerically.

Related Calculators