Derivatives of Trig Functions

When you were a kid, there were two key facts to remember, your name and your address. You
confused the two, and you might have been lost forever. You're not a kid anymore, but in this
section, there are still just two key facts to remember that you don't want to mix up, namely:

and

The derivatives of all of the other trig functions follow from these.

It is easy to get confused about which of these two derivatives has the negative sign in front. The easiest way to keep it straight is to remember,

``Sine keeps its sign, when you differentiate".

That is to say, when you differentiate the sine function, you do not change the sign for the result. I guess you could remember, ``Cosine changes sign," but it's not as catchy.

Showing that the derivative of the sine and cosine functions are what
they are by using the limit definition of the derivative is a little tricky.
It uses the fact that we already made a big deal over, namely

You should determine whether or not the instructor expects you to be able to derive trigonometric derivatives using this limit.

But anyway, once you know these derivatives, the rest are easy. Say you want to know the derivative of the tangent function. Well:

Since ,

and since ,

This derivative occurs enough that it is probably worth memorizing:

But the derivatives of and are most likely not worth memorizing, as they are easily derived. Again, this depends a lot on the professor. Make sure that you can find the derivatives of these functions using the derivatives of sine and cosine. It makes a typical test question.

And by the way, just as cosine picks up a minus sign upon differentiation, so do the other two trig functions that begin with `c', namely cosec and cot. So you can just remember, ``To avoid a grade of C-, C gets a negative".