Solving the Derivative of cos(x)

In this lesson, we will see how to find the derivative of cos(x). We will also see the derivatives of all the basic trigonometric functions and discuss why it is useful to be familiar with these derivatives.
Steps to Solve
The derivative of cos(x) is often one that people know by memory, but they don’t know how to show it. Let’s take a look at how we can find the derivative of cos(x) if we can’t remember it off the top of our heads. To do this, we’re going to make use of the fact that cos(x) = 1 / sec(x). We’re going to find the derivative of 1 / sec(x), and in doing so, we will also find the derivative of cos(x), since they’re equal.
cosderv1
Notice that 1 / sec(x) is a quotient. Therefore, we are going to use the quotient rule for derivatives to find the derivative. The quotient rule for derivatives is as follows.
cosderv2
The other facts that we will need to know to find this derivative are as follows:
The derivative of sec(x) is sec(x)tan(x)
The derivative of a constant is 0
tan(x) = sin(x) / cos(x)
(a/b) / (c/d) = (a/b) * (d/c) = (ad / bc)
Okay, now that we have a game plan and all the facts that we’re going to need, let’s dive in! The first thing we want to do is use the quotient rule on 1 / sec(x). In 1 / sec(x), the function in the numerator is f(x) = 1, and the function in the denominator is g(x) = sec(x). We plug these into the quotient rule, which you can see play out here:
cosderv3
Now we’re going to simplify. This is where our facts come into play. During the first part of simplification, we use facts 1 and 2, which you can see here:
cosderv5
And now, we can and will simplify further using facts 3 and 4, like this right here:
cosderv6
We can put all this work in a nice organized and compact form, as follows:
cosderv7
So, with all said and done, we see that the derivative of cos(x) is -sin(x).
If you want to make sure that you get all of the different steps that we’re showing, feel free to pause and take note of them. If this is just a handy review for you, let’s move on.
Trigonometric Function Derivatives
As was said, the derivative of cos(x) is one that is well known. Though it’s always useful to know how to find a derivative, knowing some key derivatives by memory is quite useful. Suppose you were solving a problem and along the way, you had to find the derivative of cos(x). If you had it memorized that the derivative of cos(x) is -sin(x), you wouldn’t have to go through all the steps we just went through. We see that knowing certain derivatives can save us a lot of time and work.

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