Question

How do you differentiate `(-2x^2 -5x-4 )/ (cos2x)` using the quotient rule?

Answer 1

`(df)/(dx)=-(cos2x(4x+5)-2sin2x(2x^2+5x+4))/(cos^2 2x)`

Explanation:

Quotient rule states if `f(x)=(g(x))/(h(x))`

then `(df)/(dx)=((dg)/(dx)xxh(x)-(dh)/(dx)xxg(x))/(h(x))^2`

Hence, as `f(x)=(-2x^2-5x-4)/(cos2x)`

`(df)/(dx)=(cos2x(-4x-5)-(-2sin2x)(-2x^2-5x-4))/(cos^2 2x)`

= `(-4xcos2x-5cos2x-2sin2x(2x^2+5x+4))/(cos^2 2x)`

= `(-4xcos2x-5cos2x-4x^2sin2x-10xsin2x-8sin2x)/(cos^2 2x)`

= `-(cos2x(4x+5)-2sin2x(2x^2+5x+4))/(cos^2 2x)`