Partial Derivative Calculator

How to Use This Calculator

1. Enter f(x,y) (e.g. x^2 + x*y + y^2).
2. Enter the x and y values at which to evaluate.
3. Select ∂f/∂x or ∂f/∂y.
4. Use the Gradient tab to get both partial derivatives and magnitude.
5. The Professional tab computes the full Hessian matrix and directional derivative.

Formula

Example

Frequently Asked Questions

• What is a partial derivative? ▼
  The partial derivative ∂f/∂x measures how f(x,y) changes when x changes and y is held constant. It is computed numerically as (f(x+h,y)−f(x−h,y))/(2h).
• What is the gradient of a function? ▼
  The gradient ∇f = (∂f/∂x, ∂f/∂y) is a vector pointing in the direction of steepest ascent. Its magnitude |∇f| is the rate of steepest increase.
• What is the Hessian matrix? ▼
  The Hessian H is the 2×2 matrix of second partial derivatives: [[fxx, fxy],[fxy, fyy]]. Its determinant det(H) = fxx·fyy − fxy² is used in the second derivative test to classify critical points.
• What is a directional derivative? ▼
  The directional derivative D_u f = ∇f · u gives the rate of change of f in the direction of unit vector u = (dx, dy). It equals |∇f|cos(θ) where θ is the angle between ∇f and u.
• Can this handle functions of 3 variables? ▼
  The current tool handles f(x,y). For f(x,y,z), use the Triple Integral Calculator tab which accepts 3-variable expressions.

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