Question

What is the derivative of `cosx^2`?

Answer 1

`-sin2x`

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)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(cosx)=-sinx)color(white)(a/a)|)))`
`color(blue)"-----------------------------------------------"`

`f(g(x))=cos^2x=(cosx)^2rArrf'(g(x))=2(cosx)^1=2cosx`

and `g(x)=cosxrArrg'(x)=-sinx`
`color(blue)"-----------------------------------------------"`
Substitute these values into (A)

`rArrf'(g(x))=2cosx(-sinx)=-2sinxcosx`

Using the following trig. identity to simplify.

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(sin2x=2sinxcosx)color(white)(a/a)|)))`

`rarrf'(g(x))=-2sinxcosx=-sin2x`