Essential Concepts
- The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
- Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.
Key Equations
- Chain rule, one independent variable
[latex]\dfrac{dz}{dt}=\dfrac{\partial z}{\partial x}\cdot\dfrac{dx}{dt}+\dfrac{\partial z}{\partial y}\cdot\dfrac{dy}{dt}[/latex] - Chain rule, two independent variables
[latex]\begin{array}{c} \dfrac{\partial z}{\partial u} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u} \\\dfrac{\partial z}{\partial v} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial v} \end{array}[/latex] - Generalized chain rule
[latex]\dfrac{\partial w}{\partial t_{j}}=\dfrac{\partial w}{\partial x_{1}}\dfrac{\partial x_{1}}{\partial t_{j}}+\dfrac{\partial w}{\partial x_{2}}\dfrac{\partial x_{1}}{\partial t_{j}}+\ldots+\dfrac{\partial w}{\partial x_{m}}\dfrac{\partial x_{m}}{\partial t_{j}}[/latex]
Glossary
- generalized chain rule
- the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables
- intermediate variable
- given a composition of functions (e.g., [latex]f\left(x(t),y(t)\right)[/latex]) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function [latex]f\left(x(t),y(t)\right)[/latex] the variables [latex]x[/latex] and [latex]y[/latex] are examples of intermediate variables
- tree diagram
- illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for