Continuously differentiable functions

Question

Let $f, g,$ be $ C^2$ functions $\mathbb{R} \rightarrow \mathbb{R}$,

$ F: \mathbb{R}^2 \rightarrow \mathbb{R}, F(x,y) = f(x+g(y))$

Check that $(D_1F)(D_{12}F)=(D_2F)(D_{11}F)$

I know how to compute first derivatives of F:

$D_1F = D_1f(x+g(y))\times 1 $

$D_2F = D_1f(x+g(y))\times g'(y)$

but I tend to get stuck on the second derivatives. I would appreciate any help!

Answer 1

So I followed the hints and this is what I got:

$ D_{11} = f''(x+g(y))$

$D_{12}=f''(x+g(y))g'(y)$

so then if we combine everything

$D_2FD_{11}= f'(x+g(y))g'(y)f''(x+g(y))$

$D_1FD_{12} = f''(x+g(y))g'(y)f'(x+g(y))$

and we see they are indeed equal.