Chain Rule

analysis

Definition

Chain Rule (Multivariable)

Let $D\subseteq {\mathbb{R}}^n$ be an open set,$f:D\to \mathbb{R}$ a scalar function, and $\mathbf{g}:\mathbb{R}\to {\mathbb{R}}^n$ a vector function defined by:

$$\mathbf{g}(x)=\left[\begin{array}{c}{g}_1(x)\\ g_2(x)\\ ⋮\\ g_n(x)\end{array}\right]$$

where $\mathbf{g}(\mathbb{R})\subseteq D$. Let $F(x)=(f\circ \mathbf{g})(x)=f(\mathbf{g}(x))$. Then, the derivative $F^{\prime }(x)$ is given by the sum of the products of the partial derivatives along all paths of dependency:

$$F^{\prime }(x)=\frac{\partial F}{\partial x}=\sum _{i=1}^n\frac{\partial f}{\partial g_i}\cdot \frac{\partial g_i}{\partial x}$$

Visual Intuition

Dependency Graph: The chain rule can be visualised as a tree of dependencies. To find the total derivative of $F$ with respect to $x$, one sums the contributions of all possible paths from $x$ to $F$.

Simple Scalar Case

For the composition of two scalar functions $y=f(u)$ and $u=g(x)$, the rule simplifies to:

$$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$