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:


 * $\displaystyle D_x \left({y \ \mathbf{z} }\right) = \frac{\mathrm d y}{\mathrm d x}\mathbf{z} + y\frac{\mathrm d \mathbf z}{\mathrm d x}$

Also see

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