Derivative of a composite function

Question

There are two functions f(x) and g(x):

f(x) and g(x)

I need to differentiate:

(a) g ∘ f using the chain rule

(b) h, where h = g ∘ f

I found the partial derivatives of f and g with respect to both variables, but I don’t know how to plug one function into another i.e. how to create g(f(x)).

Any help/hint is appreciated.

Thanks in advance.

Answer 1

The Jacobians are: $\frac{\partial f}{\partial x}=\begin{pmatrix} 1&1\\1&-1\end{pmatrix}$ and $\frac{\partial g}{\partial x}=\begin{pmatrix}2x_1& 2x_2\end{pmatrix}$.

Now we should multiply (after plugging $f$ into the Jacobian of $g$), and get: $\begin{pmatrix}4x_1&4x_2\end{pmatrix}$

As a check, note that $g\circ f(x)=2x_1^2+2x_2^2$.