Summary: Given a composite function of the form $h(x) = f(g(x))$, difficulties are often encountered in calculating the value of the $n$th derivative at some point $x = x_{0}$ when one attempts to determine whether its $n$th derivative becomes zero at this point, or attempts to find the sign of the $n$th derivative by differentiating it $n$ times and substituting $x_{0}$.{ }This present paper offers an alternative method that allows the investigation of the $n$th derivative of function $h(x)$ based on the investigation of functions $f(x)$ and $g(x)$ only.{ }Several examples are given, which implement the conclusions on the properties of the relation.