Question

What is the derivative of `f(x)=ln(x)/x` ?

Answer 1

By Quotient Rule,

`y'={1/x cdot x-lnx cdot 1}/{x^2}={1-lnx}/{x^2}`

This problem can also be solved by the Product Rule

`y'=f'(x)g(x)+f(x)g(x)`

The original function can also be rewritten using negative exponents.

`f(x)=ln(x)/x=ln(x)*x^-1`

`f'(x)=1/x*x^-1+ln(x)*-1x^-2`

`f'(x)=1/x*1/x+ln(x)*-1/x^2`

`f'(x)=1/x^2-ln(x)/x^2`

`f'(x)=(1-ln(x))/x^2`