Question

How do you differentiate `sin^2(2x)`?

Answer 1

`2sin4x`

Explanation:

differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|))) ........ (A)`

here `f(g(x))=sin^2(2x)=(sin2x)^2`

`rArrf'(g(x))=2sin2x`

Note, 2 applications of `color(blue)"chain rule"` required for g'(x)

`g(x)=sin2xrArrg'(x)=cos2x.d/dx(2x)=2cos2x`
`"------------------------------------------------------------------"`
Substitute these values into (A)

`2sin2x.2cos2x=4sin2xcos2x`

`color(orange)"Reminder" color(red)(|bar(ul(color(white)(a/a)color(black)(sin4x=2sin2xcos2x)color(white)(a/a)|)))`

`rArrd/dx(sin^2(2x))=4sin2xcos2x=2sin4x`