Partial derivatives of a function of a function

Question

I have:

$$R=x-f\bigl(y,z,g(z),h(z)\bigr)$$

and need to find the partial derivatives of R wrt $x,y,z$

$x,y,z$ are all independent variables.

$\frac{\partial R}{\partial x} = 1$

$\frac{\partial R}{\partial y} = -\frac{\partial f}{\partial y}$

What is $\frac{\partial R}{\partial z}$ ?

Answer 1

I'm pretty sure the answer is:

$\frac{\partial R}{\partial z} = -\biggl(\frac{\partial f}{\partial z}+\frac{\partial f}{\partial g}\frac{dg}{dz}+\frac{\partial f}{\partial h}\frac{dh}{dz}\bigg)$