Question

if (y = \frac{ln x}{x}), then (\frac{dy}{dx}=)
a (\frac{1}{x})
b (\frac{1}{x^{2}})
c (\frac{ln x - 1}{x^{2}})
d (\frac{1-ln x}{x^{2}})
e (\frac{1 + ln x}{x^{2}})

Explanation:

Step1: Apply quotient - rule for differentiation

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{u'v - uv'}{v^{2}}$. Here, $u = \ln x$, so $u'=\frac{1}{x}$, and $v = x$, so $v' = 1$.

Step2: Substitute $u$, $u'$, $v$, $v'$ into the quotient - rule formula

$\frac{dy}{dx}=\frac{\frac{1}{x}\cdot x-\ln x\cdot1}{x^{2}}$.

Step3: Simplify the expression

$\frac{1 - \ln x}{x^{2}}$.

Answer:

D. $\frac{1-\ln x}{x^{2}}$