Product Rule for Derivatives

Product Rule

• Suppose $f:{\mathbb{R}}^n\to {\mathbb{R}}^k$ is a Vector Space
• Let $c:{\mathbb{R}}^n\to \mathbb{R}$ be a scalar field
• Suppose both $f$ and $c$ are Differentiable at $x_0$
• Let $h(x)=c(x)f(x):{\mathbb{R}}^n\to {\mathbb{R}}^k$
• $Dh(x_0)=c(x_0)Df(x_0)+f(x_0)Dc(x_0)$

Intuition

• $Dh(x_0)$ and $Dc(x_0)Df(x_0)$ are both $n\times k$ matrix
• $f(x_0)$ is a $1\times k$ matrix
• $Dc(x_0)$ is a $n\times 1$ matrix

x\\ y \end{array}\right]$$Where: - $c(x) =xy$ - $$f(x)= \left[\begin{array}{cc} x\\ y \end{array}\right]$$