Question

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

Answer 1

The derivative is `f'(x)=(1-logx)/x^2`.

This is an example of the the Quotient Rule:

Quotient Rule .

The quotient rule states that the derivative of a function `f(x)=(u(x))/(v(x))` is:

`f'(x)=(v(x)u'(x)-u(x)v'(x))/(v(x))^2`.

To put it more concisely:

`f'(x)=(vu'-uv')/v^2`, where `u` and `v` are functions (specifically, the numerator and denominator of the original function `f(x)`).

For this specific example, we would let `u=logx` and `v=x`. Therefore `u'=1/x` and `v'=1`.

Substituting these results into the quotient rule, we find:

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

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