I am confused bz the chain rule of multivariable function. I know, that sometimes it is impossible to dot it.
For example i have the following functions:
$f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^3x_2\\x_1^2x_2 + x_1 + x_2^3\\\end{bmatrix}$
$g(u) = \begin{bmatrix} e^u \\ u^2 + u\\\end{bmatrix}$
Is it possible to take a derivative of $f(g)$ or $g(f)$. If not - why?
And in general, when can we take a derivative of such functions(multivariable) and when not?
$f:\Bbb R^2 \to \Bbb R^2$ and $g: \Bbb R \to \Bbb R^2$.
Both functions are of class $C^{\infty}$ on their domains, so we only need to know how composing the two functions makes sense. We can't compose $g$ with $f$, but we can compose $f$ with $g$ and $f \circ g: \Bbb R \to \Bbb R^2$: it is a vector function, and $(f\circ g)'(u) = J_f(g(u)) g'(u)$, where $J_f$ is the Jacobian matrix of $f$.