What is the derivative of #(sin x)^(3x)#?

1 Answer
Shaikh A.
Dec 6, 2016

#3(sinx)^(3x)(lnsinx+xcotx)#

Explanation:

#y=(sinx)^(3x)#
taking ln both sides
#lny=ln(sinx)^(3x)#
#lny=(3x)ln(sinx)#
Differentiate both sides
#(1/y)dy/dx=3lnsinx+3x(1/sinx)cosx#
#dy/dx=(3lnsinx+3xcotx)y#
putting value of y
#dy/dx=3(lnsinx+xcotx)(sinx)^(3x)#
it can be written as
#dy/dx=3(sinx)^(3x)(lnsinx+xcotx)#

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