Trigonometric Derivatives

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: