What is the derivative of #f(x)=2lnx^3#?

2 Answers
May 26, 2015

Using the chain rule, we can rename #u=x^3#, and thus, start working with #f(x)=f(u)=2ln(u)#.

The chain rule states that

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

So,

#(dy)/(du)=2*1/u=2/u#

#(du)/(dx)=3x^2#

Thus,

#(dy)/(dx)=2/u*3x^2=(6x^2)/u=(6cancel(x^2))/x^cancel(3)=color(green)(6/x)#

May 26, 2015

Alternative solution:

#f(x) = 2lnx^3=6lnx#

So,

#f'(x) = 6 * 1/x = 6/x#