Question

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

Answer 1

`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`