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:

$$\boxed{ \frac {d} {dx} (\sin x) = \cos x }$$

and

$$\boxed{ \frac {d} {dx} (\cos x) = -\sin x. }$$

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

$$\boxed{ \lim_{x \to 0} \frac { \sin x} {x} =1.}$$

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:

$$\begin{eqnarray*}\frac {d} {dx} (\tan x) &=& \frac {d} {dx} (\frac{\sin x}{\cos x}) \\ \end{eqnarray*}$$

By the quotient rule :

$$\begin{eqnarray*}\frac {d} {dx} (\frac{\sin x}{\cos x}) &=& \frac{(\cos x)(\sin ... ... x)( \sin x)}{ \cos^2x} \\ &=& \frac{\cos^2x+\sin^2x}{ \cos^2x} \end{eqnarray*}$$

Since $\sin^2x+\cos^2x=1$,

$$\begin{eqnarray*}&=& \frac{1}{\cos^2x} \end{eqnarray*}$$

and since $1/ \cos x = \sec x$,

$$= \sec^2x.$$

This derivative occurs enough that it is probably worth memorizing:

$$\boxed{ \frac {d} {dx} \tan x = \sec^2x }$$

But the derivatives of $\sec x, \csc x$ and $\cot x$ 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".