Let $y=g(f(x))=g(u)$, where $u=f(x)$, with $f$ and $g$ differentiable functions. Then

Whenever possible, it is best to express the final answer in terms of original variable $x$. While each function in the composition is differentiated, the respective points at which the functions are evaluated do not change; so we have $s^{\prime}(x)=g^{\prime}(f(x))f^{\prime}(x)$, not $g^{\prime}(x)f^{\prime}(x)$.