The Derivative of Cos

What is the derivative of Cos?

In calculus, the derivative of cos(x) is equal to -sin(x)

(d/dx)cos(x) = -sin(x)


cos'(x) = -sin(x)

So what does this mean and how do we get to that answer?

Well, since differentiation involves finding the rate of change (or gradient) of a function, finding the derivative of cos means that we are trying to determine the rate of change of the cos function with respect to x (the angle of the cos function).

This evaluation of the rate of change of the cos function with respect to the variable x is found to be negative sine of x (-sin(x)).

Next we will look at how we calculate the derivative of cos(x).

The derivative of cos(x) from first principles

To find the derivative of cos(x) (or in fact any function) it is often a good idea to start with first principles and the definition of the derivative of a function.

Recall that derivative of a function f(x) (or f'(x)) can be expressed as

Definition of the derivative functions using limits

In this case f(x), the function we want to differentiate is cos(x). So if we put f(x) = cos(x) into the definition of the derivative function we get

Putting cos(x) into the definition of the derivative function

Next, we need to use the following trigonomic identity relating to cos and the sum of two angles. You may recall (if you don’t then just accept it as fact for now)

Identity relating to cos of the sum of two angles which is used to help with the differentiation of cos. cos(m+n)=cos(m)cos(n)-sin(m)sin(n)

We can then apply this trig identity to the cos(x+h) term from above to get:

Insert the formula for cos of sum of angles into the derivative of cos

Then we can do some factorisation of the numerator by rearranging it a little and taking out a common cos(x) term:

rearrange the numerator and then factorise it to help find the derivative of cos.

Finally, we split the numerator up so that we can apply two formulas related to finding the limits of trigonometric functions.

Split the numerator up to help with applying two limit formulas later

The following two formulas relating to limits of sin and cos now come in handy. We can plug these two in and then solve:

Insert the limit formulas for sin and cos and then simplify to find that the derivative of cos is -sin

And that proves that the derivative of cos(x) is -sin(x)