The derivative of cosx. How to differentiate cosx.

The derivative of cos x is –sinx. Just remember cos goes to minus sin.

A more general and useful version of this is:

if y = cosf(x)

Then dy/dx = -f’(x)sinf(x)

Let’s take a look at some examples:

Example 1

Work out the derivative of y = cos4x.

f(x) = 4x so f ‘(x) = 4

Therefore:

dy/dx = -4sin4x

Example 2

Work out the derivative of y = cos(7x+3)

f(x) = 7x+3 so f ‘(x) = 7

Therefore:

dy/dx = -7sin(7x+3)

Example 3

Work out the derivative of y = cos(x-5)

f(x) = x-5 so f ‘(x) = 1

Therefore:

dy/dx = -sin(x-5)

Example 4

Work out the derivative of y = cosx²

f(x) = x² so f ‘(x) = 2x

Therefore:

dy/dx = -2xsin(x²)

Example 5

Work out the derivative of y = cos(x³+4x²)

f(x) = x³+4x²so f ‘(x) = 3x² + 8x

Therefore:

dy/dx = -(3x² + 8x )sin(x³ + 4x²)

Extra Tips

Don’t forget the negative sign in your answer.

Harder Example

Work out the gradient of the tangent at x = 10 on the graph of y = cos(3x).

In order to find the gradient of the tangent, all you need to do is plug in x = 10 into dy/dx.

However, first you need to find dy/dx.

f(x) = 3x so f ‘(x) =3

Therefore:

dy/dx = -3sin(3x)

Now plug in x =10

dy/dx = -3sin(3 X 10)

dy/dx = -3sin(30)

dy/dx = -1.5

So the gradient of the tangent is -1.5


More by this Author


Comments 2 comments

catman3000 profile image

catman3000 6 years ago from England, UK Author

Thanks for spotting the typing error Anthony.


Anthony Nixon 6 years ago

Hey guys, Example two is incorrect.

f(x)= cos(7*x+3)

f'(x) = -7sin(7*x+3)

NOT

the answer given, simple mistake just dropped a 3 :)

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.


    Click to Rate This Article
    working