# Derivative of e^(−x): Formula, Steps and Examples

> The derivative of e^(−x) is −e^(−x). See the chain rule steps, a check with the quotient rule, higher derivatives, a graph and examples like xe^(−x).

Source: https://practicecalculus.com/derivatives/e-to-the-negative-x
Updated: 2026-11-06

$$ \frac{d}{dx}\,e^{-x} = -e^{-x} $$

## At a glance

- **Derivative:** $-e^{-x}$
- **Second derivative:** $e^{-x}$
- **nth derivative:** $(-1)^n e^{-x}$
- **Same function as:** $\dfrac{1}{e^x}$
- **Integral version:** $\int e^{-x}\,dx = -e^{-x} + C$

## Key takeaways

- The derivative of $e^{-x}$ is $-e^{-x}$. The minus sign comes from the inside derivative of $-x$.
- The signs alternate: the $n$th derivative is $(-1)^n e^{-x}$.
- $e^{-x} = \frac{1}{e^x}$ is always positive but always decreasing, which is why its derivative is negative.
- The integral is also $-e^{-x} + C$, because differentiating $-e^{-x}$ gives back $e^{-x}$.

The **derivative of $e^{-x}$ is $-e^{-x}$**. The exponential stays the same, and the derivative of the exponent $-x$, which is $-1$, puts a minus sign in front.

$$\frac{d}{dx}\,e^{-x} = -e^{-x}$$

## Step by step with the chain rule

Think of $e^{-x}$ as $e^u$ with $u = -x$. The [chain rule](/rules/chain-rule) gives

$$\frac{d}{dx}e^{-x} = e^{-x}\cdot\frac{d}{dx}(-x) = e^{-x}\cdot(-1) = -e^{-x}$$

The outer step uses the fact that the [derivative of eˣ](/derivatives/e-to-the-x) is $e^x$.

## Check with the quotient rule

Since $e^{-x} = \frac{1}{e^x}$, the [quotient rule](/rules/quotient-rule) should agree:

$$\frac{d}{dx}\frac{1}{e^x} = \frac{0\cdot e^x - 1\cdot e^x}{(e^x)^2} = \frac{-e^x}{e^{2x}} = -e^{-x}$$

Same answer, with more work. The chain rule is the quick way.

## Graph of e^(−x) and its derivative

```graph
title: 'Graph of y = e^(−x) and its derivative y = −e^(−x)'
caption: 'The function $e^{-x}$ is always positive and always falling. Its derivative is the mirror image below the x-axis.'
x: [-2, 5]
y: [-4, 4]
functions:
  - expr: exp(-x)
    label: y = e^(−x)
  - expr: -exp(-x)
    label: y = −e^(−x) (derivative)
    dashed: true
points:
  - [0, 1, '(0, 1)']
  - [0, -1, 'slope −1']
```

The graph of $e^{-x}$ is the graph of $e^x$ reflected across the y-axis. Reflecting left to right turns every rising slope into a falling one, so the derivative is negative. As $x \to \infty$ both the function and its slope shrink toward $0$. This is the shape of exponential decay.

## Higher derivatives

Each derivative brings out another $-1$, so the signs alternate:

| Derivative | Result |
| --- | --- |
| $f'(x)$ | $-e^{-x}$ |
| $f''(x)$ | $e^{-x}$ |
| $f'''(x)$ | $-e^{-x}$ |
| $f^{(n)}(x)$ | $(-1)^n e^{-x}$ |

Since $f''(x) = f(x)$, the function $e^{-x}$ solves the differential equation $y'' = y$, just like $e^x$ does. And $f'(x) = -f(x)$ says it shrinks at a rate equal to its own size, which is the defining rule of exponential decay.

## Worked examples

**Example 1: product rule.** For $f(x) = xe^{-x}$, the [product rule](/rules/product-rule) gives

$$f'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1 - x)$$

So $f$ rises for $x < 1$, falls for $x > 1$, and has a maximum of $f(1) = \frac{1}{e} \approx 0.368$.

**Example 2: exponential decay.** A quantity $A(t) = 50e^{-0.2t}$ changes at the rate

$$A'(t) = 50\cdot(-0.2)e^{-0.2t} = -10e^{-0.2t}$$

At $t = 0$ it is falling by $10$ units per unit time.

**Example 3: a sum.** For $g(x) = e^x + e^{-x}$,

$$g'(x) = e^x - e^{-x}$$

Divide both by $2$ and you get the [derivative of cosh x](/derivatives/cosh-x), which is $\sinh x$.

## Common mistakes

1. **Dropping the minus sign.** $\frac{d}{dx}e^{-x} = e^{-x}$ is wrong. The inside derivative is $-1$.
2. **Writing $-xe^{-x - 1}$.** That uses the power rule on an exponential.
3. **Losing track of signs in higher derivatives.** Even derivatives are positive, odd ones are negative.
4. **Getting the integral sign wrong.** $\int e^{-x}\,dx = -e^{-x} + C$, not $e^{-x} + C$. See the [integral of e^(−x)](/integrals/e-to-the-negative-x).

For more chain rule practice with exponentials, try [derivatives of exponential and logarithmic functions](/practice-problems/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of e^(−x)?

The derivative of $e^{-x}$ is $-e^{-x}$.

### What is the derivative of 1/e^x?

Since $\frac{1}{e^x} = e^{-x}$, the derivative is $-e^{-x} = -\frac{1}{e^x}$.

### What is the second derivative of e^(−x)?

Differentiate $-e^{-x}$ again: $-(-e^{-x}) = e^{-x}$. The second derivative equals the original function.

### What is the derivative of x·e^(−x)?

By the product rule, $e^{-x} - xe^{-x} = e^{-x}(1 - x)$.