How do you differentiate #cos^2(x^3)#?

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Answer 1 · 2016-11-02T10:49:38

#(cos^2(x^3))'=-3x^2sin(2x^3)#

The differentiation of the given function is determined by applying chain rule

Let #u(x)=cos^2x and v(x)=x^3#
Then the given function is a composite of #u(x) and v(x)#

#color(blue)(u@v(x)=u(v(x))=cos^2(x^3))#

#color(red)((u@v(x))'=u'(v(x))=xxv'(x))#

#(cos^2(x^3))'=(cos^2)'(v(x))xx(x^3)'#

#(cos^2(x^3))'=(2(-sin(v(x)))cos(v(x))xx(3x^2)#

#(cos^2(x^3))'=-2sin(x^3)cos(x^3)xx(3x^2)#

#(cos^2(x^3))'=-color(brown)(sin(2x^3))xx(3x^2)#

#(cos^2(x^3))'=-3x^2sin(2x^3)#