I won't give the exact example but let's say I have this function $G: t ↦ F(Y(t))$
I know this to translate to $G(t) = F(Y(t))$
If $F(y)$ is some complicated equation with constants $a, b$, such as $y^3 - (a+b)/y^2 +a\log(|a+b+y|)$
and a function $Y'(t) = Y(t)/Y(t)(a+bY(t))$
Then how can I bring this all together to compute by hand a derivative for $G(t) = F(Y(t))$?
I think I first need to plug $Y(t)$ into the $y$ values of $F(y)$ like $Y(t)^3 - (a+b)/Y(t)^2 + a\log(|a+b+Y(t)|)$
That's about all I can figure out. Because I have $Y'(t)$ but not $Y(t)$. and I'm not sure how i can get $G'(t)$ when I don't have actual $t$ variables in any equation