Solving the Derivative of cos(2x)

Steps to Solve

We want to find the derivative of cos(2x). To do this, we will make use of the chain rule. The chain rule is a rule used in calculus to find derivatives of compositions of functions. As a reminder, a composition of functions is a function of a function.

Well, now we can look at our function cos(2x) a bit differently. Notice that if we let f(x) = cos(x) and g(x) = 2x, then f(g(x)) = cos(2x). We see that cos(2x) is actually a composition of functions. Ah-ha! That’s why we will make use of the chain rule to find its derivative. Okay, if we’re going to use this chain rule, it’s probably a good idea to actually know what that rule is.

The chain rule for derivatives states that to take the derivative of a composition of functions f(g(x)), we multiply the derivative of f with g(x) plugged in by the derivative of g(x).

f(g(x)) = f ‘ (g(x)) * g ‘ (x).

Good news! We already saw how to view cos(2x) as a composition of functions. We simply let f(x) = cos(x) and g(x) = 2x, then f(g(x)) = cos(2x). Perfect! Now we know we can use the chain rule to find the derivative of cos(2x) by plugging it into our formula. The only other information we need is as follows:

1. The derivative of cos(x) is -sin(x).
2. The derivative of 2x is 2.

Okay, let’s get to work!

f ‘ (g(x)) = f ‘ (2x) * g ‘ (x) = -sin(2x) * 2 = -2sin(2x)

Once we get all the information we need to find this derivative, we see that finding the derivative is actually quite simple. The derivative of cos(2x) is -2sin(2x).

Checking Your Work

Alright, we’ve seen that finding the derivative of cos(2x) isn’t so hard, but what if we want to make sure we have the right answer? More good news! To check our work, we can use integrals. Integrals are basically just derivatives in reverse. Finding the integral of a function f is the same as finding the function that f is the derivative of. We can relate integrals and derivatives in the following way.