Chain Rule for Real-Valued Functions

Theorem
Let $f: \R^n \to \R, \mathbf x \mapsto z$ be a differentiable real-valued function.

Let $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \in \R^n$.

Further, let every entry $x_i: 1 \le i \le n$ represent an implicitly defined differentiable real function of $t$.

Then $z$ is itself differentiable WRT $t$ and:

where $\dfrac {\partial z}{\partial x_i}$ is the partial derivative of $z$ WRT $x_i$.

Corollary
Let $\Psi$ represent a differentiable function of $x$ and $y$.

Let $y$ represent a differentiable function of $x$.

Then:

Proof
By hypothesis, $f$ is differentiable.

From Characterization of Differentiability:

Let $\Delta t \ne 0$ and divide both sides of the equation by $\Delta t$:

Recall that each $x_i$ was defined to be differentiable WRT $t$, that is, that each $\dfrac {\mathrm dx_i}{\mathrm dt}$ exists.

Then $\Delta x_i \to 0$ as $\Delta t \to 0$.

Therefore:

Also see

 * Chain Rule
 * Total Derivative
 * Exact Differential Equation