How does the slope of a sinusoid change with its frequency?

Question

If I increase the frequency of a sinusoid does its slope change in the same proportion no matter what the factor I change it by? Or is the change proportionally smaller when sinusoids are very low frequency (aka almost flat)?

Answer 1

A sinusoid is given by $f(t)=a\sin(\omega t+t_0)$ where $a$ is amplitude and $\omega$ is the angular frequency. The slope is given by the derivative $f'(t)=a\omega\cos(\omega t+t_0)$. Fixing the amplitude $a$ but scaling $\omega$ by a factor $k$ increases the amplitude of $f'$ by the same factor $k$.