Partial Derivative Calculator – Multivariable Calculus
Calculate partial derivatives of functions with multiple variables
Table of Contents
How to Use
- Enter your function using x, y, z as variables (e.g., x^2+y^2)
- Specify which variable to differentiate with respect to
- Optionally enter point coordinates to evaluate the derivative
- Click calculate to see the partial derivative and steps
What is a Partial Derivative?
A partial derivative measures how a function changes as one variable changes while holding all other variables constant. For a function f(x, y), the partial derivative with respect to x, written ∂f/∂x, treats y as a constant.
Partial derivatives are fundamental in multivariable calculus and are used to analyze functions of several variables, find rates of change, and optimize functions.
Notation and Symbols
- ∂f/∂x: Partial derivative of f with respect to x
- fₓ: Subscript notation for partial derivative
- ∂²f/∂x∂y: Mixed partial derivative
- ∇f: Gradient (vector of all partial derivatives)
Differentiation Rules
- Power rule: ∂/∂x(xⁿ) = n·xⁿ⁻¹
- Constant rule: ∂/∂x(c) = 0 for any constant c
- Sum rule: ∂/∂x(f + g) = ∂f/∂x + ∂g/∂x
- Product rule: ∂/∂x(f·g) = f·∂g/∂x + g·∂f/∂x
- Chain rule: ∂/∂x(f(g)) = f'(g)·∂g/∂x
Applications
- Optimization: Finding maxima and minima of multivariable functions
- Machine Learning: Gradient descent for training neural networks
- Physics: Heat equations, wave equations, fluid dynamics
- Economics: Marginal analysis, utility functions
- Engineering: Stress analysis, control systems
Frequently Asked Questions
- What's the difference between partial and ordinary derivatives?
- An ordinary derivative applies to functions of one variable. A partial derivative applies to functions of multiple variables and measures change with respect to one variable while treating others as constants.
- What is the gradient?
- The gradient ∇f is a vector containing all partial derivatives of a function. For f(x,y), the gradient is (∂f/∂x, ∂f/∂y). It points in the direction of steepest increase.
- What are mixed partial derivatives?
- Mixed partial derivatives involve differentiating with respect to different variables in sequence, like ∂²f/∂x∂y. By Clairaut's theorem, the order usually doesn't matter: ∂²f/∂x∂y = ∂²f/∂y∂x for most functions.
- How are partial derivatives used in machine learning?
- Partial derivatives are essential for gradient descent, the algorithm used to train neural networks. The gradient tells us how to adjust each parameter to minimize the loss function.