The Multivariable Chain Rule

⚡ Key Takeaways

One-line statement: The multivariable chain rule computes the derivative of a composition of functions when intermediate quantities depend on several variables, summing contributions through every dependency path. Key result in words: The total rate of change equals the sum, over all intermediate variables, of (rate of change of the output w.r.t. that intermediate) × (rate of change of that intermediate w.r.t. the input). Where it applies: Functions $\mathbb{R}^n \to \mathbb{R}^m$ composed with other multivariable functions; physics, optimization, machine learning (backpropagation), thermodynamics, differential geometry. Who/when: Rooted in Leibniz's and Newton's calculus (1670s); generalized to multiple variables through the 18th–19th century (Euler, Cauchy); formalized via the Jacobian by the late 1800s.

💡 The Core Insight

A change in an input variable propagates to the output through every chain of dependencies. To find the total derivative, you trace each path from input to output, multiply the local derivatives along that path, and add up all the paths. The "sum over paths" is the heart of the matter; the matrix-product formulation is just this sum written compactly.

DefinitionPartial and Total Derivatives

Let $z = f(x_1, \dots, x_n)$ be a differentiable function. The partial derivative $\frac{\partial f}{\partial x_i}$ measures the rate of change of $f$ as $x_i$ varies while all other $x_j$ ( $j\neq i$ ) are held fixed. If each $x_i = x_i(t)$ is a function of a single parameter $t$ , then $z$ becomes a function of $t$ alone, and its total derivative $\frac{dz}{dt}$ accounts for all the ways $t$ influences $z$ . The Jacobian matrix of a map $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ , $\mathbf{f} = (f_1, \dots, f_m)$ , is the $m \times n$ matrix

1

\quad D\mathbf{f} = \left(\frac{\partial f_i}{\partial x_j}\right)_{\substack{1 \le i \le m \\ 1 \le j \le n}}.

Derivation hint: stack the gradients $\nabla f_i$ as rows; entry $(i,j)$ is the sensitivity of output $f_i$ to input $x_j$ .

TheoremMultivariable Chain Rule — Scalar Output, Curve Input

Let $z = f(x_1, \dots, x_n)$ be differentiable at a point, and let each $x_i = x_i(t)$ be differentiable. Then the composition $z(t) = f(x_1(t), \dots, x_n(t))$ is differentiable and

2

\quad \frac{dz}{dt} = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}\,\frac{dx_i}{dt} = \nabla f \cdot \frac{d\mathbf{x}}{dt}.

Derivation hint: take the total differential $dz = \sum_i \frac{\partial f}{\partial x_i}\,dx_i$ and divide by $dt$ ; the dot product is gradient times velocity.

TheoremMultivariable Chain Rule — Several Independent Variables

Let $z = f(x_1, \dots, x_n)$ where each $x_i = x_i(t_1, \dots, t_k)$ . Then for each $j = 1, \dots, k$ ,

3

\quad \frac{\partial z}{\partial t_j} = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}\,\frac{\partial x_i}{\partial t_j}.

Derivation hint: hold all $t_\ell$ ( $\ell \neq j$ ) fixed and apply equation [2] with $t_j$ playing the role of $t$ .

TheoremGeneral Form — Jacobian Composition

Let $\mathbf{g}: \mathbb{R}^k \to \mathbb{R}^n$ be differentiable at $\mathbf{t}_0$ and $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ be differentiable at $\mathbf{g}(\mathbf{t}_0)$ . Then $\mathbf{h} = \mathbf{f}\circ \mathbf{g}: \mathbb{R}^k \to \mathbb{R}^m$ is differentiable at $\mathbf{t}_0$ and its Jacobian factors as a matrix product:

4

\quad D\mathbf{h}(\mathbf{t}_0) = D\mathbf{f}\big(\mathbf{g}(\mathbf{t}_0)\big)\,\cdot\,D\mathbf{g}(\mathbf{t}_0).

Derivation hint: the best linear approximation of a composition is the composition of best linear approximations; composing linear maps corresponds to multiplying their matrices.

ProofProof via Differentiability and Linear Approximation

We prove equation [2]; the general case [4] follows by applying it component-wise.

Step 1 — Differentiability hypothesis. Since $f$ is differentiable at $\mathbf{a} = \mathbf{x}(t_0)$ , by definition there is a remainder function $\varepsilon(\mathbf{h})$ with $\varepsilon(\mathbf{h})\to 0$ as $\mathbf{h}\to 0$ such that

5

\quad f(\mathbf{a}+\mathbf{h}) - f(\mathbf{a}) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(\mathbf{a})\,h_i + \|\mathbf{h}\|\,\varepsilon(\mathbf{h}).

Derivation hint: this is precisely the definition of (total) differentiability — linear part plus error vanishing faster than $\|\mathbf{h}\|$ .

Step 2 — Increment from $t$ . Let $\mathbf{h} = \mathbf{x}(t_0 + \Delta t) - \mathbf{x}(t_0)$ , so $h_i = x_i(t_0+\Delta t) - x_i(t_0)$ . Substituting into [5] and dividing by $\Delta t$ :

6

\quad \frac{z(t_0+\Delta t) - z(t_0)}{\Delta t} = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(\mathbf{a})\,\frac{h_i}{\Delta t} + \frac{\|\mathbf{h}\|}{\Delta t}\,\varepsilon(\mathbf{h}).

Derivation hint: divide both sides of [5] by the scalar $\Delta t$ and identify $z(t)=f(\mathbf{x}(t))$ .

Step 3 — Take the limit. As $\Delta t \to 0$ : each $\frac{h_i}{\Delta t} \to x_i'(t_0)$ by differentiability of $x_i$ ; the factor $\frac{\|\mathbf{h}\|}{\Delta t}$ stays bounded (it tends to $\|\mathbf{x}'(t_0)\|$ ); and $\varepsilon(\mathbf{h})\to 0$ since $\mathbf{h}\to 0$ . Hence the error term vanishes and

7

\quad \frac{dz}{dt}(t_0) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(\mathbf{a})\,x_i'(t_0). \qquad \blacksquare

Derivation hint: product of (something bounded) and (something $\to 0$ ) is $\to 0$ ; the surviving sum is the claim.

∎

⚠️ Why differentiability — not just partial derivatives — is required

The proof crucially uses the differentiability of $f$ (Step 1), which is stronger than mere existence of partial derivatives. A function can have all partials at a point yet fail to be differentiable, and the chain rule can then break. Always verify differentiability (e.g., continuous partials suffice — class $C^1$ ).

mathematician · Augustin-Louis Cauchy | 1789–1857

While Leibniz and Newton wielded chain-rule reasoning informally, Cauchy is central to the rigorous $\varepsilon$ – $\delta$ foundation that makes the proof above legitimate. His insistence on precise definitions of limit and derivative transformed the chain rule from a symbolic manipulation of differentials into a theorem with hypotheses (differentiability) and a remainder analysis. The Jacobian matrix formulation that crowns the modern statement is named for Carl Gustav Jacob Jacobi (1804–1851), whose 1841 treatise De determinantibus functionalibus systematized functional determinants.

🧠 Sum Over Dependency Paths

Draw a dependency graph: arrows from each input to each intermediate variable, then from each intermediate to the output. To find $\frac{\partial(\text{output})}{\partial(\text{input})}$ : 1. Find every directed path from the input node to the output node. 2. Along each path, multiply the local derivatives on its edges. 3. Sum over all paths. If $z$ depends on $u,v$ , and both $u,v$ depend on $s,t$ , there are two paths from $s$ to $z$ (through $u$ , and through $v$ ), giving $\frac{\partial z}{\partial s} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial s} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial s}$ . The single-variable rule is the degenerate case of exactly one path.

Figure

Dependency graph for

z=f(u,v)

with

u=u(s,t),\,v=v(s,t)

. Two paths carry

s

to

z

.

ExampleA particle's temperature along a path

Given: Temperature field $T(x,y) = x^2 y + 3y$ (°C, with $x,y$ in metres). A particle moves along $x(t) = \cos t$ , $y(t) = \sin t$ (metres), with $t$ in seconds. Find $\frac{dT}{dt}$ at $t = \pi/2$ s.

Step 1 — Partials.

8

\quad \frac{\partial T}{\partial x} = 2xy, \qquad \frac{\partial T}{\partial y} = x^2 + 3.

Derivation hint: differentiate $x^2y+3y$ treating the other variable as constant.

Step 2 — Velocities. $\dfrac{dx}{dt} = -\sin t$ , $\dfrac{dy}{dt} = \cos t$ .

Step 3 — Chain rule [2].

9

\quad \frac{dT}{dt} = (2xy)(-\sin t) + (x^2+3)(\cos t).

Derivation hint: substitute the partials and velocities into $\frac{dT}{dt}=T_x x' + T_y y'$ .

Step 4 — Evaluate at $t=\pi/2$ : $x=\cos(\pi/2)=0$ , $y=\sin(\pi/2)=1$ , $\sin t = 1$ , $\cos t = 0$ .

10

\quad \frac{dT}{dt} = (2\cdot0\cdot1)(-1) + (0^2+3)(0) = 0.

Derivation hint: plug numbers; both terms vanish.

\boxed{\dfrac{dT}{dt}\Big|_{t=\pi/2} = 0\ \text{°C/s}}

ExamplePolar-coordinate partial derivatives

Given: $f(x,y) = x^2 + y^2$ with $x = r\cos\theta$ , $y = r\sin\theta$ . Find $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$ at $r=2$ , $\theta = \pi/4$ .

Step 1 — Partials of $f$ . $\dfrac{\partial f}{\partial x} = 2x$ , $\dfrac{\partial f}{\partial y} = 2y$ .

Step 2 — Partials of $x,y$ .

11

\quad \frac{\partial x}{\partial r}=\cos\theta,\ \frac{\partial y}{\partial r}=\sin\theta,\ \frac{\partial x}{\partial\theta}=-r\sin\theta,\ \frac{\partial y}{\partial\theta}=r\cos\theta.

Derivation hint: differentiate $r\cos\theta$ and $r\sin\theta$ w.r.t. $r$ and $\theta$ .

Step 3 — Chain rule [3].

12

\quad \frac{\partial f}{\partial r} = 2x\cos\theta + 2y\sin\theta = 2r\cos^2\theta + 2r\sin^2\theta = 2r.

Derivation hint: substitute $x=r\cos\theta,\ y=r\sin\theta$ and use $\cos^2+\sin^2=1$ .

13

\quad \frac{\partial f}{\partial \theta} = 2x(-r\sin\theta) + 2y(r\cos\theta) = -2r^2\cos\theta\sin\theta + 2r^2\sin\theta\cos\theta = 0.

Derivation hint: the two terms cancel — consistent with $f=r^2$ being independent of $\theta$ .

Step 4 — Evaluate at $r=2$ : $\frac{\partial f}{\partial r} = 4$ , $\frac{\partial f}{\partial\theta}=0$ .

\boxed{\dfrac{\partial f}{\partial r}\Big|_{r=2} = 4,\qquad \dfrac{\partial f}{\partial\theta} = 0}

(Check: $f = r^2 \Rightarrow \partial f/\partial r = 2r = 4$ . ✓)

⚠️ Distinguish

\frac{\partial z}{\partial t}

from

\frac{dz}{dt}

When $z = f(x,t)$ has an explicit $t$ and also $x = x(t)$ , the total derivative includes both the direct and indirect dependence:

14

\quad \frac{dz}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\frac{dx}{dt}.

Derivation hint: $t$ reaches $z$ directly (through the explicit slot) and indirectly (through $x$ ); sum both paths. Forgetting the $\partial f/\partial t$ term is the single most common error in physics and economics. The notation $\frac{Df}{Dt}$ (material derivative) in fluid dynamics is precisely this distinction made explicit.

CounterexamplePartials exist but the chain rule fails

Let

15

\quad f(x,y) = \begin{cases} \dfrac{xy}{x^2+y^2} & (x,y)\neq(0,0)\\[4pt] 0 & (x,y)=(0,0).\end{cases}

Derivation hint: a classic homogeneous-degree-zero example. Both partials exist at the origin: $f_x(0,0)=f_y(0,0)=0$ (since $f\equiv0$ on both axes). Now take the path $x(t)=t$ , $y(t)=t$ . Then $f(t,t) = \frac{t^2}{2t^2}=\frac12$ for $t\neq0$ , so $z(t)=\frac12$ (constant), giving $\frac{dz}{dt}=0$ . The naive formula $f_x x' + f_y y' = 0\cdot1 + 0\cdot1 = 0$ happens to agree here — but $f$ is not even continuous at the origin (limit along $y=x$ is $\tfrac12$ , along axes is $0$ ), so it is not differentiable, and the chain rule has no theorem to stand on. The agreement is a coincidence, and along other constructions the naive formula misfires. Moral: existence of partials does not license the chain rule.

connection · Where the chain rule plugs into the wider theory

The multivariable chain rule is the computational engine behind several neighbouring results. The Gradient Vector appears as the row-vector factor $\nabla f$ in equation [2], and the rule shows $\frac{dz}{dt}=\nabla f\cdot \mathbf{v}$ , directly yielding the Directional Derivative. The general Jacobian form [4] is the multivariable face of the Single-Variable Chain Rule, with matrix multiplication replacing scalar multiplication. It underlies the Implicit Function Theorem (differentiate a constraint $F(x,y)=0$ along a curve), the change-of-variables formula in Multiple Integrals (through $\det D\mathbf{g}$ ), and the transformation of vector fields in Differential Geometry. In its iterated, layered form it is exactly the Backpropagation Algorithm.

🔬 Concrete uses across science and engineering

⏳ History

- 1676 — Gottfried Wilhelm Leibniz develops differential notation

dy

,

dx

and manipulates differentials of composite quantities, implicitly using chain-rule reasoning.

- 1690s — Johann and Jakob Bernoulli apply chain-rule techniques to curves and the brachistochrone, treating differentials of functions of functions.

- 1755 — Leonhard Euler, in Institutiones calculi differentialis, systematizes partial differentiation and differentials of functions of several variables.

- 1797 — Joseph-Louis Lagrange introduces functional/prime notation, recasting the chain rule symbolically.

- 1821 — Augustin-Louis Cauchy's Cours d'analyse supplies rigorous limit definitions, enabling a proper proof with error terms.

- 1841 — Carl Gustav Jacob Jacobi publishes De determinantibus functionalibus, establishing the Jacobian determinant and matrix as the organizing object for transformations.

- late 1800s — The matrix-product form of the chain rule [4] becomes standard as linear algebra and vector analysis mature (Gibbs, Heaviside).

- 1960s–1986 — Reverse-mode automatic differentiation is developed (Linnainmaa, 1970) and popularized as backpropagation (Rumelhart, Hinton, Williams, 1986), recognizing the chain rule as the foundation of neural-network training.

open-problem · Edge cases and ongoing care

The clean theorem [4] requires differentiability at the relevant points; weakening to mere directional or partial differentiability invalidates it (see the counterexample). Subtle questions persist in: (a) non-smooth analysis — chain rules for subdifferentials and Clarke generalized gradients of nonsmooth functions require careful constraint qualifications, and a single clean equation does not always hold; (b) infinite dimensions — in Banach/Fréchet spaces the chain rule for Fréchet derivatives holds, but Gâteaux differentiability is too weak, mirroring the finite-dimensional cautions; (c) automatic differentiation correctness — efficiently and exactly differentiating programs with control flow, recursion, and nondifferentiable kinks (ReLU, $\max$ ) is an active engineering and theoretical area, where the naive chain rule can yield wrong or undefined values at kinks. The conceptual rule is settled; its rigorous reach into nonsmooth and infinite-dimensional settings remains a refined subject.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also