Unleash the Power of the Chain Rule

Effortlessly compute derivatives of composite multivariable functions with our interactive calculator.

Multivariable Chain Rule Calculator

Outer Function (e.g., f(u, v) = u^2 + v)

Inner Functions (e.g., u(x, y) = x*y; v(x, y) = x+y)

Variables to Differentiate By (e.g., x, y)

Derivative:

Calculation Steps:

Understanding the Multivariable Chain Rule

The chain rule in multivariable calculus is used to find the derivatives of composite functions of several variables. It's an extension of the single-variable chain rule, allowing us to differentiate functions where variables depend on other variables.

Formula: If f is a function of variables u, v, ..., and u, v, ... are functions of x, y, ..., then the partial derivative of f with respect to, say, x is given by:

$$ \frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x} + ... $$

How to Use This Calculator:

Enter the 'Outer Function' in terms of intermediate variables (e.g., u, v).
Enter the 'Inner Functions' defining how these intermediate variables depend on the final variables (e.g., u(x, y) = ..., v(x, y) = ...), separated by semicolons.
Specify the 'Variables' with respect to which you want to find the partial derivatives, separated by commas.
Click 'Calculate Derivative' to compute and view the results and step-by-step breakdown.

This tool uses math.js for mathematical computations and MathJax for formula rendering.