How do you find the derivative of #cos^2x sinx#?

2 Answers
Nov 27, 2016

#"the derivative of y="cos^2 x sin x" is "y^'=-2cos x+3cos^3 x#

Explanation:

#a=cos ^2x#

#b=sin x#

#y=cos^2 x sin x#

#y=a*b#

#y^'=a^'*b+b^'*a#

#a^'=-2cos x*sin x#

#b^'=cos x#

#y^'=-2cos x*sin x* sin x+cos x*cos^2 x#

#y^'=-2cos x*sin^2 x+cos^3 x#

#sin^2 x=1-cos^2 x#

#y^'=-2cos x(1-cos^2 x)+cos^3 x#

#y^'=-2cos x+2cos^3 x+cos^3 x#

#y^'=-2cos x+3cos^3 x#

Nov 27, 2016

I would replace #cos^2x# by #1-sin^2x#

Explanation:

#cos^2xsinx = (1-sin^2x)sinx = sinx-sin^3x#

The derivative is #cosx-3sin^2xcosx#.

There are other ways to write the derivative.