Question

How do you take the derivative of `y = sin x cos x - cos^2x`?

Answer 1

`(dy)/(dx)=cos2x+sin2x`

Explanation:

Here,
`y=sinxcosx-cos^2x`

Using Product Rule and Chain Rule ,

`dy/dx=sinxd/(dx)(cosx)+cosxd/(dx)(sinx)-2cosxd/(dx)(cosx)`

`=sinx(-sinx)+cosxcosx-2cosx(-sinx)`

`=-sin^2x+cos^2x+2sinxcosx`

`=cos^2x-sin^2x+2sinxcosx`

`=cos2x+sin2x`

Note :

`(1) sin2x=2sinxcosx`

`(2)cos2x=cos^2x-sin^2x`