Given $F(x,y,z) = f(g(x+y),h(y+z))$ what is the partial derivatives of $F$ using partial derivative of $f,g,h$ for $x,y,z$?
I don't have any clue, i used to solve when i know the function and not in the abstract form.
Edit : Let $f(u,v)$ and so :
$\frac{\partial F(x,y,z)}{\partial x} = \frac{\partial f(g(x+y),h(y+z))}{\partial u} \frac{d g(x+y)}{d x} = \frac{\partial f(g(x+y),h(y+z))}{\partial u} g'(x+y)$
$\frac{\partial F(x,y,z)}{\partial y} = \frac{\partial f(g(x+y),h(y+z))}{\partial u} \frac{d g(x+y)}{d y} + \frac{\partial f(g(x+y),h(y+z))}{\partial u} \frac{d h(y+z)}{d y } = \frac{\partial f(g(x+y),h(y+z))}{\partial u} g'(x+y)+ \frac{\partial f(g(x+y),h(y+z))}{\partial u} h'(y+z) $
$\frac{\partial F(x,y,z)}{\partial z} = \frac{\partial f(g(x+y),h(y+z))}{\partial u} \frac{d h(y+z)}{d z} = \frac{\partial f(g(x+y),h(y+z))}{\partial u} h'(y+z) $
Is this answer correct ?