Derivative of outer product of two vectors

Question

Given two vectors $u(x)$ and $v(x)$ for $x\in\mathbb{R}^n$ and $u(x), v(x)\in\mathbb{R}^m$ what is the derivative of their outer product? $$ \frac{d}{dx} u(x) v(x)^\top = ? $$ Perhaps it is this? $$ \frac{d u(x)}{dx} v(x)^\top + u(x)\left(\frac{d v(x)}{dx}\right)^\top $$

Answer 1

Hint

Writing $u(x)=(u_1(x),\cdots, u_m(x))$ (similar for $v(x)$),

$$\big(u(x)v(x)^T\big)_{ij}= u_i(x)v_j(x).$$

Edit

$$(u_iv_j)'=u'_iv_j+u_iv'_j,$$

and thus $$(uv^T)'=u'v^T+u(v')^T.$$