What is the derivative of $lnx /x$ ?

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Answer 1 · 2018-04-08T23:56:10

$d/dxlnx/x=1/x^2(1-lnx)$

We could use the Quotient Rule, but it's nice to avoid it where it is unnecessary. Instead, we can rewrite as

$x^-1lnx$ and differentiate with the Product Rule, recalling that $d/dxlnx=1/x=x^-1$

$d/dxlnx/x=x^-1(d/dxlnx)+lnx(d/dxx^-1)$

$=x^-1x^-1-x^-2lnx$

$x^-1x^-1=x^(-1-1)=x^-2=1/x^2$

Thus,

$d/dxlnx/x=1/x^2-lnx/x^2$

$d/dxlnx/x=1/x^2(1-lnx)$

What is the derivative of lnx /xlnx /x?