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.
$\endgroup$2 Answers
$\begingroup$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
$\nabla (fg) = g\nabla f + f \nabla g. \tag3$
$\endgroup$ $\begingroup$Yes you can. Gradient is a vector of derivatives with respect to each component of vector x, and for each the product is simply differentiated as usual.
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