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

Answer 1

Emily Ammerman

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

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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Answer 2

Samantha Adkison

The first derivative of ( f(x) = \frac{\ln(x-2)}{x-2} ) is ( f'(x) = \frac{1}{x(x-2)} ). The second derivative is ( f''(x) = -\frac{1}{x^2} + \frac{1}{(x-2)^2} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.