Partial Derivative Calculator
Find df/dx, df/dy, and the gradient magnitude of any f(x,y).
x^3*y^2 + 2*x*y, sin(x)*y, x^2 + y^2.Operators:
* for multiply, ^ for power.Partial Derivative Calculator: Multi-Variable Calculus
The Partial Derivative Calculator computes df/dx and df/dy of a multi-variable function f(x, y) with respect to each variable independently. When taking a partial derivative, all other variables are treated as constants — isolating the rate of change in just one dimension at a time. This is the foundation of multi-variable calculus, optimization, and machine learning.
The Gradient Vector
Together, the partial derivatives form the gradient vector: ∇f = (df/dx, df/dy). The gradient points in the direction of steepest ascent on the function's surface. Its magnitude ‖∇f‖ = √[(df/dx)² + (df/dy)²] quantifies the steepness. Gradient descent — the learning algorithm behind neural networks — moves opposite the gradient to minimize loss.
Applications
- Thermodynamics: Partial derivatives express how pressure, volume, and temperature relate at constant conditions.
- Economics: Marginal product of capital = df/dK holding labor L constant.
- Computer Graphics: Normal vectors on 3D surfaces are computed from partial derivatives of surface equations.
- Fluid Mechanics: Navier-Stokes equations governing fluid flow consist entirely of partial derivatives of velocity and pressure.
Numerical Method
We use the central-difference formula df/dx ≈ [f(x+h,y) − f(x-h,y)] / (2h) with h = 1×10-7, achieving O(h²) accuracy — highly reliable for smooth functions.