# 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