AP Calculus AB and BC

Derivative of e^x/x: Answer, Proof, Mistakes

Q: What is the derivative of e^x/x?

It is \frac{e^x(x - 1)}{x^2}, from the quotient rule.

Q: Where is e^x/x smallest?

At x = 1 on the positive branch, where the value is e. The derivative changes from negative to positive there, so it is a minimum.

The derivative of e^x/x with respect to x is e^x(x - 1)/x^2. The quotient rule gives x times e^x minus e^x, all over x squared, and factoring e^x out of the numerator leaves e^x(x - 1)/x^2. The formula holds for every x except 0, where the original function is undefined.

\frac{d}{dx}\left[\frac{e^x}{x}\right] = \frac{e^x(x-1)}{x^2}

The proof: quotient rule, then factor e^x

With $f(x) = e^x$ on top and $g(x) = x$ underneath, $f'(x) = e^x$ and $g'(x) = 1$ .

\frac{d}{dx}\left(\frac{e^x}{x}\right) = \frac{e^x\cdot x - e^x\cdot 1}{x^2} = \frac{e^x(x - 1)}{x^2}

Both numerator terms carry a factor of $e^x$ , because $e^x$ is its own derivative (Topic 2.7). Factoring it out is the last step graders expect, and it is what makes the sign of the derivative readable.

What the factored form tells you

$e^x > 0$ for every $x$ , and $x^2 > 0$ for every $x \neq 0$ , so the sign of the derivative is the sign of $x - 1$ by itself.

The function decreases on $x < 0$ and on $0 < x < 1$ , then increases for $x > 1$ . On the branch $x > 0$ that makes $x = 1$ a minimum.

\frac{e^1}{1} = e \approx 2.718

There is a vertical asymptote at $x = 0$ . The numerator approaches 1 while the denominator approaches 0, so the quotient runs to $+\infty$ from the right and to $-\infty$ from the left.

Common mistakes

Answering $e^x$ by differentiating the top and the bottom separately.
Reversing the numerator to $\frac{e^x(1 - x)}{x^2}$ , which flips the sign everywhere.
Cancelling an $x$ from $x - 1$ against the $x^2$ . The $x - 1$ is a difference, not a factor of $x$ , so nothing cancels.
Rewriting as $e^x x^{-1}$ and then keeping only one product rule term. Both survive: $e^x x^{-1} - e^x x^{-2}$ , which factors straight back to $\frac{e^x(x - 1)}{x^2}$ .

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of e^x/x?

It is $\frac{e^x(x - 1)}{x^2}$ , from the quotient rule.

Where is e^x/x smallest?

At $x = 1$ on the positive branch, where the value is $e$ . The derivative changes from negative to positive there, so it is a minimum.

What is the integral of e^x/x?

It has no elementary antiderivative. Mathematicians name it the exponential integral $\mathrm{Ei}(x)$ , and AP Calculus never asks you to find it, even though the derivative is routine.