Question

what is the derivative for lnx/x ?

Answer 1

We can do quotient rule or product rule for this, whicever works easier for you. If I do quotient rule I would do this:

\[\frac{ f'(x)g(x) - f(x)g'(x) }{ [g(x)]^{2} }\] Following that, I get:
\[\frac{ \frac{ 1 }{ x }*x- lnx(1) }{ x^{2} } \implies \frac{ 1-lnx }{ x^{2} }\]

Answer 2

If we were to do the product rule way, although not really necessary (I know some people like doing product rule(, then we would need to move the x to the numerator as an x^(-1) and then we would follow:

\[f'(x)g(x) + f(x)g'(x)\]
\[\frac{d}{dx}lnx(x^{-1})= \frac{ 1 }{ x }(x^{-1}) + lnx(-1)x^{-2} \implies \frac{ 1 }{ x^{2} }-\frac{ lnx }{ x^{2} }\]