How do you differentiate #f(x)=ln(x^2)# using the chain rule?

1 Answer
Ken C.
May 29, 2016

Take the derivative of #x^2# and divide it by #x^2# to get #f'(x)=2/x#.

Explanation:

The chain rule tells us that the derivative of a compound function - like #ln(x^2)#, which is made up of two functions (#lnx# and #x^2#), is the derivative of the whole thing times the derivative of the inside function. In math terms:
#f(g(x))'=f'(g(x))*g'(x)#

As it applies to the natural log function, the chain rule says:
#ln(u)'=u'*1/u=(u')/u#
Where #u# is a function of #x#.

In our case (#f(x)=ln(x^2)#), #u=x^2#, so the derivative is:
#f'(x)=(x^2')/(x^2)=(2x)/x^2=2/x#

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