Question

Find the derivative of​

Answer 1

Explanation:

`\boxed{d((u)/(v))=(u'v-uv')/(v^2) }`

`Let\ u=2cos^2((x)/(2) )-1`

`v=cos^2(x)`

Then

`u'=2(2cos((x)/(2)))(-sin((x)/(2) ))((1)/(2) )`

`=-2sin((x)/(2)) cos((x)/(2) )`

`=-sin(x)`

`v'=2(cos(x))`

`=2cos(x)`

`d((2cos^2((x)/(2) )-1)/(cos^2x) ) = (-sin\ x(cos^2x)-(2cos^2((x)/(2) )-1)(2cos\ x))/((cos^2x)^2)`

`= (-sin(x) cos^2(x)-(4cos^2((x)/(2))cos(x) -2cos(x)))/(cos^4(x))`

`= (-sin(x) cos^2(x)-4cos^2((x)/(2))cos(x) +2cos(x))/(cos^4(x))`

`= (cos(x)(-sin(x) cos(x)-4cos^2((x)/(2)) +cos(x)))/(cos^4(x))`

`= (-sin(x) cos(x)-4cos^2((x)/(2)) +cos(x))/(cos^3(x))`