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How do you differentiate #f(x)=cos^2x#?

Answer 1

#f'(x)=-sin2x#

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

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(2/2)|)))#

#"Express " f(x)=cos^2x=(cosx)^2#

#"let "u=cosxrArr(du)/(dx)=-sinx#

#"then "y=u^2rArr(dy)/(du)=2u#

#rArrdy/dx=2u(-sinx)#

Rewrite u in terms of x.

#rArrdy/dx=-2sinxcosx#

#color(orange)"Reminder" sin2x=2sinxcosx#

#rArrdy/dx=-sin2x#

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Answer 2

Christopher Allers

#dy/dx = -sin2x#

#f(x)= cos^2x = (cosx)^2#

Using the chain rule

let #u = cosx# then #y= u^2#

#(du)/dx = -sinx# then #dy/(du) = 2u#

Therefore #dy/dx = (du)/dx . dy/(du)#

#dy/dx=(-sinx)(2u)#

But #u = cosx#

#(-sinx)(2cosx)= -2sinxcosx#

Reminder #-2sinxcosx = -sin2x#

#dy/dx = -sin2x#

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Answer 3

Ezra Adams

To differentiate ( f(x) = \cos^2(x) ), you can use the chain rule. The derivative is ( f'(x) = -2\cos(x) \sin(x) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.