Are there cases where using the definition is easier than using the derivative to study the monotonic behaviour of a function?

Question

This is more a curiosity,

Say I have a function $f \in C^{1}(A)$ where $A \subset \mathbb{R}$. What could be an instance of function where studying the monotonic behaviour using the definition is better than using the derivative? Any example which is not trivial would be appreciated.

Answer 1

I think in the case when you have composition of incrasing functions or something similar. For example $$f(x) = e^{\arctan e^{x^{7677677}}}$$