Chapter 2:

Trig Derivatives

At about this point, you should look at a trigonometry review, such as Chapters 6 - 8 of my PreCalculus class, or David Joyce's Short Trig Course.

Several times, already, I've claimed that the derivative of the sine function is the cosine function.  It's about time to prove this, and the proof will rely heavily on the fact that we use RADIANS.  That's why, in a calculus class, we always switch everything to radian measure.

First, let's do a trace of the derivative of the sine function, using GeoGebra, and see what it looks like:

Applet:

Notice that the picture also matches up with what we know about the relationship between a function and its derivative.

So now, let's prove what seems apparent from the graph:

D_x[sin(x)] = cos(x).

This proof will need two extra bits of knowledge:

A)  The formula for the sine of the sum of angles, found here:

B)  That strange sin(x)/x limit we looked at earlier, found here:

PROOF:

Once we have this derivative, we can find the derivative of the cosine in a similar fashion.  Then we can use the definitions of the other trig. functions, along with the Quotient Rule, to find their derivatives.  You should, once you've derived them once or twice, MEMORIZE the following table:

Examples (several)

Next, The Chain Rule: