Key Equations

• Chain rule, one independent variable
  $\dfrac{dz}{dt}=\dfrac{\partial z}{\partial x}\cdot\dfrac{dx}{dt}+\dfrac{\partial z}{\partial y}\cdot\dfrac{dy}{dt}$
• Chain rule, two independent variables
  $\begin{array}{c} \dfrac{\partial z}{\partial u} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u} \\\dfrac{\partial z}{\partial v} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial v} \end{array}$
• Generalized chain rule
  $\dfrac{\partial w}{\partial t_{j}}=\dfrac{\partial w}{\partial x_{1}}\dfrac{\partial x_{1}}{\partial t_{j}}+\dfrac{\partial w}{\partial x_{2}}\dfrac{\partial x_{1}}{\partial t_{j}}+\ldots+\dfrac{\partial w}{\partial x_{m}}\dfrac{\partial x_{m}}{\partial t_{j}}$

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., $f\left(x(t),y(t)\right)$) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function $f\left(x(t),y(t)\right)$ the variables $x$ and $y$ are examples of intermediate variables

tree diagram

illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for