Derivative of Product of Real Function and Vector-Valued Function

Theorem
Let:
 * $\mathbf r:x \mapsto \mathbf z$

be a differentiable vector-valued function, where:


 * $\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix}$

such that:


 * $z_1, z_2, \cdots, z_n$

are (images of) differentiable real functions.

Let:


 * $f: x \mapsto y$

be a differentiable real function.

Then:


 * $D_x \left({y \, \mathbf z}\right) = \dfrac {\d y} {\d x} \mathbf z + y \dfrac {\d \mathbf z} {\d x}$

Also see

 * Derivative of Vector Cross Product of Vector-Valued Functions
 * Derivative of Dot Product of Vector-Valued Functions