Partial Derivative Calculator
Use x, y, z as variables. Supported operations: +, -, *, /, ^, sin(), cos(), tan(), ln(), exp(), sqrt()
What is a Partial Derivative?
A partial derivative measures how a multivariable function changes when only one of its variables changes. Unlike ordinary derivatives that deal with functions of a single variable, partial derivatives are used for functions with multiple independent variables.
The partial derivative of a function f(x, y, z, ...) with respect to x is written as ∂f/∂x, using the "∂" symbol (called "del" or "partial") to distinguish it from the total derivative d/dx.
How to Calculate Partial Derivatives
To calculate a partial derivative with respect to a specific variable:
- Treat all other variables as constants
- Apply the standard differentiation rules to the variable of interest
- Simplify the resulting expression
For example, if f(x, y) = x²y + sin(y), then ∂f/∂x = 2xy (treating y as a constant) and ∂f/∂y = x² + cos(y) (treating x as a constant).
Applications of Partial Derivatives
Partial derivatives have numerous applications across various fields:
- In physics, for describing rates of change in multiple dimensions
- In economics, for analyzing marginal changes in multivariable models
- In optimization problems, for finding maxima and minima of multivariable functions
- In thermodynamics, for relating various physical properties
- In machine learning, for gradient-based optimization algorithms
Higher-Order Partial Derivatives
Just as with ordinary derivatives, you can take partial derivatives multiple times. These are called higher-order partial derivatives.
For example, ∂²f/∂x² represents taking the partial derivative with respect to x twice, while ∂²f/∂y∂x means first differentiating with respect to x, then with respect to y.