assuming differentiability of functionsthe substitution t=g(x,y) converts F(t) into f(x,y) where f(x,y)=F[(g(x,y))]

Show that

$\displaystyle \frac{\partial f}{\partial x}=F'[(g(x,y))]\frac{\partial g}{\partial x}, \frac{\partial f}{\partial y}=F'[(g(x,y))]\frac{\partial g}{\partial y}$

am I supposed to use a first order Taylor function with error term?

for instance h(a+y)-h(a)=f[g(a+y)]-f[g(a)]=f(b+v)-f(b) can be used to prove that h is differentiable at a (I guess it means that the limit exists) and the total derivative h'(a) is equal to the composition f'(b)=g'(a)