3295263856329

18 More
about Differentiation
The previous chapter showed that successive differentiation of sine waves results in the
same equation with its starting point shifted. Starting at sin x, successive derivatives are
cos x, sin x, cos x, sin x, cos x, sin x, . . ., cycling through these four over and over.
It is particularly easy with x in radians. If the angle is in degrees, it has to be converted
into radians for this method to work. Sometimes the quantity is not itself an angle, but
is something that trigonometry can represent as an angle.
FREQUENCY AND PERIOD OF A SINE WAVE
Suppose you are analyzing waves and you come across the equation y sin 2x instead
of y sin x. Now consider this:
y dy sin 2(x dx)
sin 2x cos 2 dx cos 2x sin 2 dx
sin 2x 2 dx cos 2x
If you let y sin 2x in the last expression above, then you can substitute, subtract y
from each side, and finally divide each side by dx, getting
y dy y 2 dx cos 2x
dy 2 dx cos 2x
dy/dx 2 cos 2x
Next, take y sin 3x. The derivative can be found in the same way. You get
dy/dx 3 cos 3x
251
Copyright © 2008, 1999 by The McGraw-Hill Companies, Inc. Click here for terms of use.