Calculus – Product Rule for Gradient

Question

I have two functions scalar functions of vector $\textbf{x}$, $f(\textbf{x})$ and $g(\textbf{x})$. I know the gradient of $f(\textbf{x})$ (i.e. $\triangledown f(\textbf{x})$) and I want to find the gradient of $f(\textbf{x})g(\textbf{x})$. Can I use the product rule $$\triangledown f(\textbf{x})g(\textbf{x})=g(\textbf{x})\triangledown f(\textbf{x})+f(\textbf{x})\triangledown g(\textbf{x}).$$
I mean does the product rule of differentiation also apply to gradients? Thanks in advance.

Answer 1

Yes, the product rule as you have written it applies to gradients. This is easy to see by evaluating $\nabla (fg)$ in a Cartesian system, where

$(\nabla f)_i = \dfrac{\partial f}{\partial x_i}; \tag 1$

then we have

$(\nabla (fg))_i = \dfrac{\partial (fg)}{\partial x_i} = \dfrac{\partial f}{\partial x_i}g + f\dfrac{\partial g}{\partial x_i} = g(\nabla f)_i + f(\nabla g)_i; \tag 2$

since (2) holds for each coordinate variable $x_i$, we have

$\nabla (fg) = g\nabla f + f \nabla g. \tag3$