Cálculo
Multivariable Chain Rule Calculator
Calculate partial derivatives of composite multivariable functions using the chain rule. Visualize step-by-step derivatives. Ideal for calculus students and professionals.
Multivariable Chain Rule Calculator
Derivative:
Calculation Steps:
Derivative with respect to (∂f/∂):
- $$ \frac{\partial f}{\partial <span x-text="step.outerVar"></span>} = <span x-text="step.partialOuterExpr"></span> $$ , $$ \frac{\partial <span x-text="step.outerVar"></span>}{\partial <span x-text="stepResult.variable"></span>} = <span x-text="step.partialInnerExpr"></span> $$ , Term: $$ (\frac{\partial f}{\partial <span x-text="step.outerVar"></span>}) (\frac{\partial <span x-text="step.outerVar"></span>}{\partial <span x-text="stepResult.variable"></span>}) = <span x-text="step.termExpr"></span> $$
- Final Partial Derivative: $$ \frac{\partial f}{\partial <span x-text="stepResult.variable"></span>} = <span x-text="stepResult.derivativeExpr"></span> $$
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.