- Education and Science
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:
Work out the derivative of y = cos4x.
f(x) = 4x so f ‘(x) = 4
dy/dx = -4sin4x
Work out the derivative of y = cos(7x+3)
f(x) = 7x+3 so f ‘(x) = 7
dy/dx = -7sin(7x+3)
Work out the derivative of y = cos(x-5)
f(x) = x-5 so f ‘(x) = 1
dy/dx = -sin(x-5)
Work out the derivative of y = cosx²
f(x) = x² so f ‘(x) = 2x
dy/dx = -2xsin(x²)
Work out the derivative of y = cos(x³+4x²)
f(x) = x³+4x²so f ‘(x) = 3x² + 8x
dy/dx = -(3x² + 8x )sin(x³ + 4x²)
Don’t forget the negative sign in your answer.
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
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