Derivative of `ln(x)/x`

Step-by-step differentiation

Answer

\frac{d}{d x} [\frac{ln ( x )}{x}]

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

Step-by-step solution

1Quotient Rule
$\frac{d}{d x} [\frac{ln ( x )}{x}] = \frac{\frac{d}{d x} [ ln ( x ) ] \cdot x - ln ( x ) \cdot \frac{d}{d x} [ x ]}{( x ) ^{2}}$
Let $f = ln (x)$ and $g = x$ . The quotient rule: $(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - f \cdot g ^{'}}{g ^{2}}$ .
↳Find f′: Differentiate the numerator, $f = ln (x)$ :
2Derivative of ln
$\frac{d}{d x} [ln (x)] = \frac{1}{x}$
The derivative of $ln (x)$ is $\frac{1}{x}$ — a fundamental logarithm rule.
↳Find g′: Differentiate the denominator, $g = x$ :
3Power Rule
$\frac{d}{d x} [x] = 1$
$x$ is $x^{1}$ : by the power rule, the exponent $1$ comes down and the power drops to $x^{0} = 1$ , so the derivative is $1$ .
=Combine
$\frac{d}{d x} [\frac{ln ( x )}{x}] = \frac{\frac{1}{x} \cdot x - ln ( x ) \cdot 1}{( x ) ^{2}} = \frac{1 - ln ( x )}{x ^{2}}$
With $f^{'} = \frac{1}{x}$ and $g^{'} = 1$ , substitute into the quotient rule and simplify.
✓Final Answer
$\frac{d}{d x} [\frac{ln ( x )}{x}] = \frac{1 - ln ( x )}{x ^{2}}$
The first derivative simplifies to $\frac{1 - ln ( x )}{x ^{2}}$ .

Understanding this derivative

Differentiating ln(x)/x exercises 3 distinct techniques — the quotient rule, the power rule and the natural log rule. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The division calls for the quotient rule: the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all over the denominator squared. The order of subtraction in the numerator matters — swapping it flips the sign of the answer. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. The natural logarithm differentiates to the reciprocal: d/dx[ln x] = 1/x. When the argument of the log is itself a function, the chain rule divides its derivative by that argument.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of ln(x)/x

Step-by-step solution

Understanding this derivative

Derivative of `ln(x)/x`