Chain Rule for Multivariable Functions

Definition

The multivariable chain rule calculates the derivative of a composite function by summing the contributions of change from each intermediate variable.

Why It Matters

It allows for the calculation of complex, multi-layered rates of change, which is vital for understanding how multiple shifting variables interact in a single system.

Core Concepts

• Case 1 (One parameter): If $w = f(x, y)$ and $x=x(t), y=y(t)$: $\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt}$
  • How to read: “The derivative d w d t equals the partial derivative of w with respect to x times the derivative d x d t, plus the partial derivative of w with respect to y times the derivative d y d t.”
  • Meaning / when to use: $w$ changes with $t$ only through $x(t)$ and $y(t)$. Add the rate of change along each intermediate path—this is the dot product $\nabla f \cdot \mathbf{r}'(t)$.
• Case 2 (Several independent parameters): If $x=g(r, s), y=h(r, s)$: $\frac{\partial w}{\partial r} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial r}$
  • How to read: “The partial derivative of w with respect to r equals the partial derivative of w with respect to x times the partial derivative of x with respect to r, plus the partial derivative of w with respect to y times the partial derivative of y with respect to r.”
  • Meaning: Holding $s$ fixed, track how $w$ responds to $r$ through both $x$ and $y$. $\frac{\partial w}{\partial s} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial s}$
  • How to read: “The partial derivative of w with respect to s equals the partial derivative of w with respect to x times the partial derivative of x with respect to s, plus the partial derivative of w with respect to y times the partial derivative of y with respect to s.”
  • Meaning: Same logic with $r$ held fixed—every dependency path from $s$ to $w$ must be summed.
• Tree Diagrams: A visual tool used to track dependencies and ensure every “path” of influence is accounted for.

Connected Concepts

• The Chain Rule
• Determinants and Cramer’s Rule
• The LIATE Rule