(image taken from Karl's Calculus (

Karl's Calculus Tutor - 4.3 Derivatives: More Rules to Live By)

There are 2 wavy lines both starting from origin (lets call the wavy line that starts at the upper part TOP and the lower line BOTTOM) following the red line y=x. One is the inverse of the other. ( Remember that inverses are reflected off the y=x line).

At the same time, there is a sinosoidal curve which is supposed to be the derivative (the rate) of one of the wavy line TOP.

Mr Karl has plotted the derivative of the other wavy line (the BOTTOM) as a series of spikes, reflecting the fact that $\displaystyle ({1 \over 8})^{-1} = 8$.

But is this characteristic of all derivatives inverses? Wont the derivative be more correct if its just another sinusoidal curve (phased shifted by half a cycle) instead of the spikes?

Please comment.