What is the derivative of y ＝ ln x x ?

Question

What is the derivative of       y    ＝                  ln        x            x      ?

Nick Osborn · Accepted Answer

Use the quotient rule, which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}] = \frac{f' (x) g (x) - g' (x) f (x)}{{[g (x)]}^{2}}

Applying this to

y = \frac{\ln x}{x}

, we see that

y' = \frac{x \frac{d}{d x} [\ln x] - \ln x \frac{d}{d x} [x]}{x^{2}}

Since

\frac{d}{d x} [\ln x] = \frac{1}{x}

and

\frac{d}{d x} [x] = 1

,

y' = \frac{x (\frac{1}{x}) - \ln x}{x^{2}}

y' = \frac{1 - \ln x}{x^{2}}

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