The official definition for the word wave function is "a function that satisfies a wave equation and describes the properties of a wave."

Square the wave function.

The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.

A wave function is normalized by determining normalization constants such that both the value and first derivatives of each segment of the wave function match at their intersections.

If instead you meant renormalization, that is a different problem having to do with elimination of infinities in certain wave functions.

A simple wave function can be expressed as a trigonometric function of either sine or cosine.

lamba = A sine(a+bt) or lamba = A cosine(a+bt)

where

lamba = the y value of the wave

A= magnitude of the wave

a= phase angle

b= frequency.

the derivative of sine is cosine and the derivative of cosine is -sine

so

the derivative of a sine wave function would be y'=Ab cosine(a+bt)

""""""""""""""""""" cosine wave function would be y' =-Ab sine(a+bt)