Partial Derivative Calculator

Compute partial derivatives of multivariable functions with respect to any variable. This calculator provides step-by-step solutions for first and second order partial derivatives, helping you understand the differentiation process in multivariable calculus.

Function f(x,y,z,...):

With respect to variable:

Order of derivative:

Evaluate at point (optional):

About Partial Derivatives

A partial derivative of a multivariable function is its derivative with respect to one variable while holding the other variables constant.

First order: ∂f/∂x measures how f changes as x changes, holding y constant
Second order: ∂²f/∂x² measures the rate of change of the first derivative

Result

Enter a function and click "Calculate" to see the result.

Solution Steps

Solution steps will appear here after calculation.

Graph visualization will appear here for functions of 1-2 variables.

Understanding Partial Derivatives

Partial derivatives are fundamental in multivariable calculus, used extensively in physics, engineering, economics, and machine learning. They measure how a function changes as one of its input variables changes, while keeping all other input variables constant.

First Order Partial Derivatives

The first order partial derivative of a function f(x,y) with respect to x is denoted ∂f/∂x and represents the rate of change of f in the x-direction. Similarly, ∂f/∂y represents the rate of change in the y-direction.

For example, for f(x,y) = x²y + sin(y):

∂f/∂x = 2xy (treat y as constant)
∂f/∂y = x² + cos(y) (treat x as constant)

Second Order Partial Derivatives

Second order partial derivatives measure how the first derivatives change. There are four possible second derivatives for a function of two variables:

∂²f/∂x² - derivative of ∂f/∂x with respect to x again
∂²f/∂y² - derivative of ∂f/∂y with respect to y again
∂²f/∂x∂y - derivative of ∂f/∂x with respect to y
∂²f/∂y∂x - derivative of ∂f/∂y with respect to x

Applications of Partial Derivatives

Gradient and Optimization

Partial derivatives form the gradient vector ∇f, which points in the direction of steepest ascent. This is crucial in optimization problems and machine learning algorithms like gradient descent.

Physics and Engineering

Partial differential equations describe phenomena like heat conduction, wave propagation, and fluid dynamics. The heat equation ∂u/∂t = α∇²u relates temperature changes over time to spatial variations.

Economics

In economics, partial derivatives represent marginal concepts - the marginal product of labor is the partial derivative of the production function with respect to labor input.