Partial Derivative Calculator

Understanding Partial Derivatives

A partial derivative measures how a multivariable function changes as one variable varies while holding others constant. It's fundamental in physics, engineering, economics, and machine learning. For a function \( f(x,y) \), the partial derivative with respect to \( x \) is denoted \( \frac{\partial f}{\partial x} \).

How to Use This Tool

Enter a function of at least two variables (x and y). Use standard math notation: ^ for powers, * for multiplication, sin, cos, exp, ln. Our solver parses and differentiates symbolically to provide step-by-step reasoning and the final derivative expression.

Common use cases:

• Gradient vector computation for optimization problems
• Tangent planes and linear approximations in 3D space
• Higher-order derivatives and Hessian matrix for critical point analysis
• Economics: Marginal utility and partial elasticities
• Physics: Partial differential equations (heat, wave, Laplace)

Example functions you can try:

• f(x,y) = x^2*y + sin(x*y) → ∂f/∂x = 2xy + y*cos(xy)
• f(x,y) = e^(x*y) + ln(x+y) → ∂f/∂y = x*e^(xy) + 1/(x+y)
• f(x,y) = 3*x^2*y^3 → ∂²f/∂x² = 6*y^3

Why use our calculator? It delivers exact symbolic results, shows derivation steps, and works on any device — from smartphones to desktops. Bookmark this page for quick access during homework or research.