Partial differentiation of implicit function $y = x-z(f,g,h)$

Question

I need to figure out how to find $\frac {\partial Y}{\partial f}$ and $\frac {\partial Y}{\partial g}$.

The implicit function is $Y = x - z(f,g,h)$.

Right now I think the answer is something like $\frac {\partial Y}{\partial f} = -z'(f,g,h) \cdot \frac{dz}{df}$ and $$\frac {\partial Y}{\partial g} = -z'(f,g,h) \cdot \frac{dz}{dg}$$ but I'm not sure if that is correct. Any tips would be appreciated.

Answer 1

Assuming that $x$ is not a function of $f$, then $\frac{\partial y}{\partial f} = -\frac{\partial z}{\partial f}$, there's no $z'$ to speak of.

On the other hand, if $f = f(w)$ (for example), then $\frac{\partial y}{\partial w} = -\frac{\partial z}{\partial f}\frac{df}{dw}$ by the chain rule.