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

1 Answer
Topscooter
Dec 27, 2015

#f'(x) = -ln(x-2)/(x-2)^2# and #f''(x) = (1-2ln(x-2))/(x-2)^3 #

Explanation:

This is a quotien, so we apply the quotient rule here to have the first derivative of this function.

#f'(x) = (1/(x-2)*(x-2) - ln(x-2))*1/(x-2)^2 = -ln(x-2)/(x-2)^2#.

We do it again in order to have the 2nd derivative of the function.

#f''(x) = (1/(x-2)*(x-2)^2 - ln(x-2)(2(x-2)))*1/(x-2)^4 =((x-2) - 2ln(x-2)(x-2))/(x-2)^4 = (1-2ln(x-2))/(x-2)^3#

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