I have struggle determining the jacobian and the hessian of the following functions: $h(x) := a(x) \cdot g(x)$ and $h(x) := f(x) A g(x)$ with $a : \mathbb{R}^{m} \rightarrow \mathbb{R}$, $f,g : \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ and $A \in \mathbb{R}^{n \times n}$. Can somebody tell me a general approach with such composite functions? I know that you can somehow apply the product rule here but neither on the internet nor in my analysis 2 book I found a proof for the multivariate product rule or a general definition. I know this is a relatively basic question. Many thanks!
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The general definition is here: https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative But sometimes it is easier to just compute all the partials seperately. – Mason Nov 03 '22 at 19:19
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See Product Rule for vector output functions. If you want to systematically learn differential calculus in several variables in a general setting, then take a look at the Loomis and Sternberg reference I mention. See also Deriving successively $\dot{y}=f(y)$ or second order derivative of $F(t)$ for some examples of computing higher derivatives (though not directly for your given function). – peek-a-boo Nov 03 '22 at 22:03