What are the first and second derivatives of #f(x)=ln((x-1)^2/(x+3))^(1/3) #?

1 Answer
Mar 7, 2016

#1/3[ln(x-1)^2 -ln(x+3)]=1/3[2ln(x-1)-ln(x+3)]=2/3 ln(x-1)-1/3ln(x+3)#
#[f'(x)=2/(3(x-1) ) -1/(3(x+3))]->[f''=-2/(3(x-1)^2)+1/(3(x+3)^2)]#


First use the properties of logarithms to simplify. Bring the exponent to the front and recall that the log of a quotient is the difference of the logs so once I dissolve it into simple logarithmic form then I find the derivatives. Once I have the first derivative then I bring up the #(x-1)# and# (x+3)# to the top and apply power rule to find the second derivative. Note that you can use chain rule as well but simplifying might be a bit harder and longer.