I have learned for the total derivative: $$\frac{dF(x(z),y(z))}{dz} = \frac{\partial F(x,y)}{\partial x} \frac{dx}{dz} + \frac{\partial F(x,y)}{\partial y} \frac{dy}{dz}.$$
Now I need the partial derivative: $$ \frac{\partial F(\phi_1(x,z),\phi_2(y,z))}{\partial z} .$$
Is it correct that: $$ \frac{\partial F(\phi_1(x,z),\phi_2(y,z))}{\partial z} = \frac{\partial F(\phi_1,\phi_2)}{\partial \phi_1} \frac{\partial \phi_1}{\partial z} + \frac{\partial F(\phi_1,\phi_2)}{\partial \phi_2} \frac{\partial \phi_2}{\partial z} ?$$
I am confused because of the difference between total and partial derivative.
If $x$ and $y$ are independent from variable z then your result is correct.
If $x=x(z)$ and $y=y(z)$, then it should be
$$ \frac{\partial F(\phi_1(x,z),\phi_2(y,z))}{\partial z} = \frac{\partial F}{\partial \phi_1} \left( \frac{\partial \phi_1}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial \phi_1}{\partial z} \right) + \frac{\partial F}{\partial \phi_2} \left( \frac{\partial \phi_1}{\partial y}\frac{\partial y}{\partial z}+ \frac{\partial \phi_2}{\partial z} \right) $$