Chain Rule

Summary

The chain rule differentiates compositions. In several variables it multiplies Jacobian factors along each path of dependence.

Prerequisites

Derivatives, Partial Derivatives

Main Result / Formula

One variable

If $y = f (u)$ and $u = g (x)$ , then

\frac{d y}{d x} = f^{'} (u) g^{'} (x) .

Two variables, one independent

If $z = f (x, y)$ and $x = x (t)$ , $y = y (t)$ , then

\frac{d z}{d t} = f_{x} \frac{d x}{d t} + f_{y} \frac{d y}{d t} .

Special case $z = f (x, y)$ with $x = g (y)$ :

\frac{d z}{d y} = f_{x} (g (y), y) g^{'} (y) + f_{y} (g (y), y) .

Implicit differentiation form

If $F (x, y) = c$ defines $y = y (x)$ with $F_{y} \neq 0$ , then

\frac{d y}{d x} = - \frac{F_{x}}{F_{y}} .

This is not a formula for $\partial z / \partial y$ of an unconstrained $z = f (x, y)$ .

Worked Example

Let $z = x^{2} + y^{2}$ and $x = t$ , $y = t^{2}$ . Then

\frac{d z}{d t} = 2 x \cdot 1 + 2 y \cdot 2 t = 2 t + 4 t^{3} .

Common Mistakes

Writing $\partial z / \partial y = (\partial z / \partial x) (\partial x / \partial y)$ and dropping the direct $f_{y}$ term.
Mixing the implicit formula $- F_{x} / F_{y}$ into unconstrained partial derivatives.

Connections

Related: Implicit Differentiation, Directional Derivative, Chain Rules

References

The multivariable chain rule is standard calculus.^[1]

OpenStax, Calculus Volume 3, https://openstax.org/details/books/calculus-volume-3 ↩︎