The Chain Rule and Directional Derivatives

The multivariable chain rule generalizes the single-variable version to compositions of multivariable functions, while directional derivatives measure rates of change along arbitrary directions.

The Multivariable Chain Rule

Theorem8.2Multivariable Chain Rule

Let $\mathbf{g} : \mathbb{R}^m \to \mathbb{R}^n$ be differentiable at $\mathbf{a}$ and $f : \mathbb{R}^n \to \mathbb{R}^p$ be differentiable at $\mathbf{g}(\mathbf{a})$ . Then the composition $f \circ \mathbf{g}$ is differentiable at $\mathbf{a}$ and $D(f \circ \mathbf{g})(\mathbf{a}) = Df(\mathbf{g}(\mathbf{a})) \circ D\mathbf{g}(\mathbf{a})$ In matrix form, this is the product of Jacobian matrices: $J_{f \circ g}(\mathbf{a}) = J_f(\mathbf{g}(\mathbf{a})) \cdot J_g(\mathbf{a})$ .

ExampleChain rule with parametric curves

If $f(x, y)$ is differentiable and $x = x(t)$ , $y = y(t)$ are differentiable functions of $t$ , then $\frac{d}{dt}f(x(t), y(t)) = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$ For instance, if $f(x,y) = x^2 y$ and $x = \cos t$ , $y = \sin t$ , then $\frac{d}{dt}f = 2xy(-\sin t) + x^2(\cos t) = -2\cos t \sin^2 t + \cos^3 t$ .

Directional Derivatives

Definition

The directional derivative of $f : \mathbb{R}^n \to \mathbb{R}$ at $\mathbf{a}$ in the direction of a unit vector $\mathbf{u}$ is $D_\mathbf{u} f(\mathbf{a}) = \lim_{h \to 0} \frac{f(\mathbf{a} + h\mathbf{u}) - f(\mathbf{a})}{h} = \nabla f(\mathbf{a}) \cdot \mathbf{u}$ when $f$ is differentiable at $\mathbf{a}$ . The maximum value of $D_\mathbf{u} f$ is $\|\nabla f(\mathbf{a})\|$ , achieved when $\mathbf{u} = \nabla f(\mathbf{a}) / \|\nabla f(\mathbf{a})\|$ .

The Jacobian Matrix

Definition

For a function $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^m$ with components $f_1, \ldots, f_m$ , the Jacobian matrix at $\mathbf{a}$ is $J_\mathbf{f}(\mathbf{a}) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}$ The Jacobian matrix represents the total derivative $D\mathbf{f}(\mathbf{a})$ as a linear map $\mathbb{R}^n \to \mathbb{R}^m$ .

RemarkGeometric meaning of the Jacobian

The Jacobian matrix describes how the function $\mathbf{f}$ locally distorts space near $\mathbf{a}$ . Its determinant (when $m = n$ ) measures the factor by which volumes are scaled, which explains the role of the Jacobian determinant in the change of variables formula for multiple integrals.