What is the derivative of cosx^2?

1 Answer
Jul 11, 2016

-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