In the last section we discussed the chain rule. In this section we are going to learn to take the derivatives of some of the more common trigonometric functions.The following gives a summary of the these derivatives:

As you can see the sine and cosine functions are derivatives of each other. We can take the derivative an infinite number of times.

Let's try to work a problem involving trigonometric derivatives. Suppose we had: and we wanted to take the derivative of f(x).

The first thing to notice is that this function is a product, and when we want to take the derivative of a product we have have to use the product rule. The product rule looks like this:

Applying it to our function we get:

That's all there is to taking the derivative of a trigonometric function. Let's try another problem. Suppose we had a function that looked like this:. How will we take the derivative?

This is a good example of the chain rule. If you don't remember the chain rule it looks like this: Remember the bold x stands for any mathematical expression not just the single variable x.

Applying it to our equation we get something that simplifies to the following:

That wasn't so bad. However we can try something tricky. Suppose we wanted to find the equation of the tangent to f(x) at x = / 2.

First we need to find the slope of the line at / 2. We have the derivative of f(x); we just need to plug in the / 2 into it. If you're not sure about taking the derivative you can refresh your memory here. The slope turns out to be 0 because the cosine function is 0 at ninety degrees. So the slope on the tangent line is 0(it's a perfectly horizontal line.) Now we need to find the point at which the line runs through f(x). Plugging / 2 into f(x), we get 1. So the tangent line runs through the point ( / 2, 1) and the equation for this line is y = 0 * x + 1. We can check our answer with a graph: