$\frac{d}{dy} F(g(y),y) = ?$

Question

Given that we know if the integral of $f(y)$ is $F(y)$ then we can say that $\frac{d}{dy} F(y) = F'(y) = f(y)$.

But what does $\frac{d}{dy} F(g(y),y)$ equal to? Can we say that it is $f(g(y),y)?$

Answer 1

Since $F$ is a function of two variables, some caution is required. If $F$ depends on two variables say $w$ and $z$, i.e., $F=F(w,z)$ then $$\frac{d}{dy}F(g(y),y)=\frac{\partial F}{\partial w}(g(y),y)\cdot g'(y)+\frac{\partial F}{\partial z}(g(y),y) \cdot 1.$$