Prove chain rule in vector notation

Question

Please, help me with prove the following:

Prove gradient composition rule for the function f(x) = h(g(x))

h: R $\rightarrow$ R, g: $R^n$ $\rightarrow$ R

$\triangledown_x$ f(x) = h'(g(x))$\triangledown_x$g(x)

using vector notations. I know how to do it through limits, but I have no idea how to solve it using vectors.

Answer 1

You can derive this by considering one component at a time.

Let's illustrate this with $n=2$, say with $\mathbf{x}=[x,y]$. Then $\nabla_\mathbf{x}$ becomes $\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right]$.

Now by the scalar (i.e. 'normal') chain rule, treating y as a constant, $$\frac{\partial f}{\partial x}(x,y) = h'(g(x,y)) \frac{\partial g}{\partial x}(x,y)$$ and treating x as a constant, $$\frac{\partial f}{\partial y}(x,y) = h'(g(x,y)) \frac{\partial g}{\partial y}(x,y)$$

Answer 2

For $x=(x_1,...,x_n)$ we have

$f(x)=f(x_1,....,x_n)=h(g(x_1,...,x_n))$. With the one - dimensional chain rule we get

$f_{x_i}(x_1,....,x_n)=h(g(x_1,...,x_n))g_{x_i}(x_1,....,x_n)$ , $i=1,2,..,n$.

Can you proceed ?