How to find the derivative of cos^2x?

Question

I seem to be confused.
I was told to use the chain rule, but I don't see how it can apply.
Can't I just substitute (cos x)(cos x) in its place and then use the product rule? Isn't cos^2 x = (cos x)(cos x)?

Please let me know if I am right. If not, then please explain how to use the chain rule in this situation. Thank you.

How are you getting:
-2sinxcosx = -sin2x
?

Answer 1

cos^2x is the same as writing (cosx)^2, so we do need the chain rule
derivative of (cosx)^2 would be as follows:
= 2cosx * -sinx [note: d/dxcosx = -sinx]
= -2sinxcosx = -sin2x

You could also do it your way:
(cos x)(cos x)
So using the product rule, you'll still get
-sinxcosx + -sinxcosx = -2sinxcosx = -sin2x... either method works

***COMMENT TO YOUR ADDITIONAL DETAIL:
There is a trig identity that states that sin2x = 2sinxcosx so I just replaced 2sinxcosx with sin2x, and since there was a negative sign, it would be -sin2x

Answer 2

Derivative Of Cos 2x

Answer 3

This Site Might Help You.

RE:
How to find the derivative of cos^2x?
I seem to be confused. I was told to use the chain rule, but I don't see how it can apply. Can't I just substitute (cos x)(cos x) in its place and then use the product rule? Isn't cos^2 x = (cos x)(cos x)? Please let me know if I am right. If not, then please explain how to use the chain rule in...

Answer 4

Cos^2x= (1+cos2x)/2
so
d/dx ( cos^2x) = d/dx (1/2) + d/dx(cos2x/2)
= 0 + (-sin2x) 2/2
= - sin2x

Answer 5

y = cosx ^2
dy/dx = 2*cosx * d/dx (cosx)
= 2*cosx * -sinx
= -2sinx cosx
= -sin(2x)

Method 2:
y = cos(x) * cos(x) = uv
dy/dx = u dv/dx + v du/dx
= cos(x) * -sin(x) + cos(x) * -sin(x)
= same as above

Method 3:
y = cos^2 (x) = 1/2 + 1/2 cos(2x)
dy/dx = 0 + 1/2 * 2 * -sin(2x)
= -sin(2x)

Answer 6

Look at the beautifully typeset PDF for the solution. It looks as it would written in a textbook.
http://www.tomsmath.com/derivative-of-cosine-squared.html

Answer 7

Method 1
y = (cos x) ²
dy/dx = 2 (cos x )(- sin x)
dy/dx = - 2 sin x cos x
dy/dx = - sin 2x

Method 2
y = (cos x) (cos x)
dy/dx = (- sin x)(cos x) + (- sin x) (cos x)
dy/dx = - 2 sin x cos x
dy/dx = - sin 2x

Answer 8

cos^2x=(cosx)^2
use chain rule
y=(cosx)^2
y=k^2
k=cosx
dy/dx=dy/dk *dk/dx
=2k*(-sinx)
=2(cosx)(-sinx)
=-2cosxsinx