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

1 Answer

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#.