---
title: "The Multivariable Chain Rule"
description: "Differentiate a composition when several inputs each depend on other variables, by summing one contribution per path through the dependency diagram."
canonical: https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ
source: https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ.md
retrieved: 2026-09-12
---

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# The Multivariable Chain Rule

Differentiate a composition when several inputs each depend on other variables, by summing one contribution per path through the dependency diagram.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ

## Ready when they can

- Draw the dependency diagram and read the terms off it
- Differentiate z = f(x, y) with x and y both functions of t
- Handle two intermediate variables and two independent ones

## Lesson: One term per path

The multivariable chain rule says every route from the outer variable to t contributes one product term. Draw the dependency diagram first: z on top, x and y in the middle, t at the bottom. Each path gives an outer partial times an inner derivative, and you add the paths.

**Example.** Take z = x squared + y squared with x = t and y = 2t. Substituting gives z = 5t squared, so dz/dt = 10t, which is 10 at t = 1. Or take z = 3x + 4y with x = t squared and y = t: dz/dt = 3(2t) + 4 = 6t + 4, which is 16 at t = 2.

With two independent variables you repeat the same diagram once per target. For z = f(x, y) with x(s, t) and y(s, t), hold t fixed: dz/ds = (dz/dx)(dx/ds) + (dz/dy)(dy/ds). Same picture, new target at the bottom.

**Tip.** The single variable chain rule is the one path case: derivative of the outside times derivative of the inside. Every multivariable term has exactly this shape. If a tree feels messy, shrink it to this case first.

**Recap.** Draw the diagram, write one product per path, and add them.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Tangent Planes and Linear Approximation](https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m)
- [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn)

## Opens up

- [Backpropagation: Assigning Blame for an Error](https://lightmysky.com/learn/computing/backpropagation-assigning-blame-for-an-error-mt_5bwaSWfF96)
- [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad)