How to represent y as a function of w? Assume : $F(y)=G(w)$ where $F,G$ are two real-valued functions from $R \to R$.

We want to find the function $C(w)$ such that : $F'(y)=C(w)$ and C should be built based on F and G.

Thanks so much.
Define

$$ H(y,w) \equiv F(y)-G(w) $$

and notice that $H(y,w)=0$. Then, assuming F and G are continuously differentiable and $G'(w) \neq 0$, it follows from the Implicit Function Theorem that

$$ \frac{dw}{dy} = -\frac{H_y(y,w)}{H_w(y,w)} = -\frac{F'(y)}{-G'(w)}$$

and, therefore

$$ F'(y) = G'(w)\frac{dw}{dy}.$$
Define

$$ H(y,w) \equiv F(y)-G(w) $$

and notice that $H(y,w)=0$. Then, assuming F and G are continuously differentiable and $G'(w) \neq 0$, it follows from the Implicit Function Theorem that

$$ \frac{dw}{dy} = -\frac{H_y(y,w)}{H_w(y,w)} = -\frac{F'(y)}{-G'(w)}$$

and, therefore

$$ F'(y) = G'(w)\frac{dw}{dy}.$$