AP Calculus AB and BC

Derivative of x/e^x: Answer and Maximum

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

It is \frac{1-x}{e^{x}}, equivalently (1-x)e^{-x}.

Q: Where is the maximum?

At x = 1, with value \frac{1}{e}.

Q: Why rewrite it as x e^-x?

It converts a quotient rule into a product rule and removes the cancellation step at the end.

The derivative of x over e to the x is 1 minus x, all over e to the x. Rewriting the expression as x times e to the negative x turns a quotient rule problem into a product rule problem, which is less error prone.

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

Rewrite before differentiating

A negative exponent turns the quotient into a product, and the product rule is shorter here.

\frac{x}{e^{x}} = xe^{-x} \implies \frac{d}{dx} = e^{-x} - xe^{-x} = (1-x)e^{-x}

The quotient rule gives the same thing after cancelling a factor of $e^{x}$ , which is exactly the extra work the rewrite avoids.

The maximum at x = 1

Since $e^{-x}$ is always positive, the sign is decided by $1-x$ : positive for $x < 1$ and negative after. So there is a maximum at $x = 1$ with value $\frac{1}{e} \approx 0.368$ .

Beyond that the function decays toward $0$ , because the exponential eventually beats the linear numerator. That is the growth ordering in one picture.

Common mistakes

Answering $\frac{1}{e^{x}}$ , differentiating only the numerator.
Losing the minus sign from the chain rule on $e^{-x}$ .
Assuming the function grows because the numerator does. It peaks at $x = 1$ and falls.

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 x/e^x?

It is $\frac{1-x}{e^{x}}$ , equivalently $(1-x)e^{-x}$ .

Where is the maximum?

At $x = 1$ , with value $\frac{1}{e}$ .

Why rewrite it as x e^-x?

It converts a quotient rule into a product rule and removes the cancellation step at the end.