Derivative of `exp(x)/x`

Step-by-step differentiation

Answer

\frac{d}{d x} [\frac{e x p ( x )}{x}]

(\frac{1}{x} - (\frac{1}{x})^{2}) \cdot exp (x)

Step-by-step solution

1Quotient Rule
$\frac{d}{d x} [\frac{exp ( x )}{x}] = \frac{\frac{d}{d x} [ exp ( x ) ] \cdot x - exp ( x ) \cdot \frac{d}{d x} [ x ]}{( x ) ^{2}}$
Let $f = exp (x)$ and $g = x$ . The quotient rule: $(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - f \cdot g ^{'}}{g ^{2}}$ .
↳Find f′: Differentiate the numerator, $f = exp (x)$ :
2Derivative of eˣ
$\frac{d}{d x} [exp (x)] = e^{x}$
$e^{x}$ is its own derivative — a unique property of the natural exponential.
↳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
$\frac{d}{d x} [\frac{exp ( x )}{x}] = \frac{exp ( x ) \cdot x - exp ( x ) \cdot 1}{( x ) ^{2}} = (\frac{1}{x} - (\frac{1}{x})^{2}) \cdot exp (x)$
With $f^{'} = exp (x)$ and $g^{'} = 1$ , substitute into the quotient rule and simplify.
✓Final Answer
$\frac{d}{d x} [\frac{exp ( x )}{x}] = (\frac{1}{x} - (\frac{1}{x})^{2}) \cdot exp (x)$
The first derivative simplifies to $(\frac{1}{x} - (\frac{1}{x})^{2}) \cdot exp (x)$ .

Understanding this derivative

Differentiating exp(x)/x exercises 3 distinct techniques — the quotient rule, the power rule and the exponential 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 exponential function eˣ is its own derivative — the only function (up to constant multiples) with that property — which is why it appears unchanged in the result.

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 exp(x)/x

Step-by-step solution

Understanding this derivative

Derivative of `exp(x)/x`