How to differentiate a function of the kind $f(x,g(x))$ for $g\in C^1(\Bbb R^m)$ and $f\in C^1(\Bbb R^m\times\Bbb R^m,\Bbb R^m)$?

Question

How to differentiate a function of the kind $f(x,g(x))$ for $g\in C^1(\Bbb R^m)$ and $f\in C^1(\Bbb R^m\times\Bbb R^m,\Bbb R^m)$?

I dont see clearly how to differentiate symbolically a function with this form, how to setup the chain rule or so (if possible). Can some one enlighten me? Thank you in advance.

Answer 1

Since this involves a two-variable function $f$ you'll need partial derivatives. Let $\partial_1 f$ be the partial derivative of $f$ wrt its first argument and $\partial_2 f$ wrt its second. Then the chain rule generalizes to$$ \frac{d}{dx} f(h(x),g(x)) = \partial_1f(h(x),g(x))h'(x) + \partial_2 f(h(x),g(x))g'(x).$$