Show that $||\nabla h(x,y,z) ||^2 = 4g(x,y,z)[f'(g(x,y,z)]^2$

Question

Let $f$ be a function having a derivative $f'$ at each point in $\mathbb{R}^1$ and let $g$ be defined on $\mathbb{R}^3$ by the equation $g(x,y,x)=x^2+y^2+z^2$ and $h=f o g$.

Show that $||\nabla h(x,y,z) ||^2 = 4g(x,y,z)[f'(g(x,y,z)]^2$

Now, we know $|| \nabla h(x,y,z) ||^2 = (\frac{\partial h}{\partial x})^2 + (\frac{\partial h}{\partial y})^2+(\frac{\partial h}{\partial z})^2$ So, we need to calculate $\frac{\partial h}{\partial x}$ and others individually by the chain rule. But I can't bring the exact expression.

Answer 1

We have $h(x,y,z)=f(x^2+y^2+z^2).$ Then:

$$h_x(x,y,z))=f'(g(x,y,z))2x$$

hence

$$h_x(x,y,z))^2=[f'(g(x,y,z))]^24x^2.$$

Now compute $h_y^2$ and $h_z^2$ and add.