What is the derivative of cosx^2?
1 Answer
Jul 11, 2016
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