# Exponential Functions bˣ: Graphs, Growth, Derivative and Integral

> Exponential functions f(x) = b^x: growth and decay graphs, properties, the derivative b^x ln b, the integral b^x / ln b + C and worked examples.

Source: https://practicecalculus.com/functions/exponential-functions
Updated: 2026-11-03

$$ f(x) = b^x, \quad b > 0,\ b \ne 1 $$

## At a glance

- **Domain:** $(-\infty, \infty)$
- **Range:** $(0, \infty)$
- **Derivative:** $b^x \ln b$
- **Integral:** $\frac{b^x}{\ln b} + C$
- **Growth or decay:** Growth if $b > 1$, decay if $0 < b < 1$
- **Inverse:** $\log_b x$

## Key takeaways

- An exponential function $b^x$ has a fixed positive base $b \ne 1$ and the variable in the exponent.
- Its derivative is $b^x\ln b$: the same function, scaled by the natural log of the base.
- Its integral is $\frac{b^x}{\ln b} + C$.
- Every exponential can be rewritten as $b^x = e^{x\ln b}$, which is how both rules are derived.

An **exponential function** has a constant base raised to a variable power, $f(x) = b^x$. Each time $x$ increases by $1$, the output is multiplied by the same factor $b$, which is why exponentials model compound interest, population growth, half-lives and anything else that changes by a fixed percentage.

## Formula

$$f(x) = b^x, \qquad b > 0,\ b \ne 1$$

More generally $f(x) = A\,b^x$, where $A = f(0)$ is the starting value. The restrictions on $b$ matter: $b = 1$ gives the constant $1$, and a negative base such as $(-2)^x$ is not real for most fractional $x$. The most important base is $e$; see the [natural exponential function](/functions/natural-exponential-function).

## Graphs of exponential functions

```graph
title: 'Graphs of the exponential functions y = 2ˣ and y = (1/2)ˣ'
caption: 'Both pass through $(0, 1)$ and have the horizontal asymptote $y = 0$. Base $2$ shows growth; base $\tfrac{1}{2}$ shows decay and is the mirror image across the $y$-axis.'
x: [-4, 4]
y: [-1, 9]
functions:
  - expr: 2^x
    label: y = 2ˣ
  - expr: (0.5)^x
    label: y = (1/2)ˣ
points:
  - [0, 1, '(0, 1)']
```

## Key properties

- **Domain** all reals, **range** $(0, \infty)$; the graph always passes through $(0, 1)$.
- **Growth ($b > 1$):** increasing, $b^x \to \infty$ as $x \to \infty$ and $b^x \to 0$ as $x \to -\infty$.
- **Decay ($0 < b < 1$):** decreasing, with the opposite limits.
- **Horizontal asymptote** $y = 0$ in both cases. No $x$-intercepts.
- **Concave up** for every valid base.
- **Inverse** is the [logarithmic function](/functions/logarithmic-functions) $\log_b x$.
- **Change of base:** $b^x = e^{x\ln b}$.
- **Exponent rules:** $b^{m}b^{n} = b^{m+n}$, $\frac{b^m}{b^n} = b^{m-n}$, $(b^m)^n = b^{mn}$, $b^{-n} = \frac{1}{b^n}$.

## Derivative of bˣ

Rewrite with base $e$ and use the [chain rule](/rules/chain-rule):

$$\frac{d}{dx}b^x = \frac{d}{dx}e^{x\ln b} = e^{x\ln b}\ln b = b^x\ln b$$

The derivative is proportional to the function itself. For $b > 1$, $\ln b > 0$ (growth); for $0 < b < 1$, $\ln b < 0$ (decay). With an inner function, $\frac{d}{dx}b^{g(x)} = b^{g(x)}\ln b \cdot g'(x)$.

## Integral of bˣ

$$\int b^x\,dx = \frac{b^x}{\ln b} + C$$

Differentiate the right side to check: $\frac{b^x\ln b}{\ln b} = b^x$.

## Worked examples

**Example 1.** $\frac{d}{dx}5^x = 5^x\ln 5$.

**Example 2: chain rule.** $\frac{d}{dx}3^{x^2} = 3^{x^2}\ln 3 \cdot 2x$.

**Example 3: definite integral.** $\int_0^2 2^x\,dx = \left[\frac{2^x}{\ln 2}\right]_0^2 = \frac{4 - 1}{\ln 2} = \frac{3}{\ln 2} \approx 4.33$.

**Example 4: half-life.** A sample with a half-life of $5$ years has $A(t) = A_0\left(\frac{1}{2}\right)^{t/5}$. The decay rate is

$$A'(t) = A_0\left(\tfrac{1}{2}\right)^{t/5}\ln\tfrac{1}{2} \cdot \tfrac{1}{5} = -\frac{\ln 2}{5}A(t)$$

so the sample loses about $13.9\%$ of its current amount per year at every instant.

**Example 5: compound interest.** At $5\%$ yearly interest, $P(t) = P_0(1.05)^t$. Compounded continuously it becomes $P_0e^{0.05t}$.

## Common mistakes

1. **Using the power rule.** $\frac{d}{dx}2^x \ne x\,2^{x-1}$.
2. **Forgetting the $\ln b$ factor**, or writing $\log_{10} b$ instead of $\ln b$.
3. **Multiplying instead of dividing when integrating.** $\int 2^x\,dx = \frac{2^x}{\ln 2} + C$, not $2^x\ln 2 + C$.
4. **Writing $2 \cdot 3^x = 6^x$.** The coefficient does not combine with the base.

## How it connects

Exponentials outgrow every polynomial, a fact that drives many [limits at infinity](/blog/limits-at-infinity) and [L'Hôpital's rule](/blog/lhopitals-rule) problems. Geometric sequences are exponentials sampled at whole numbers, which leads to [geometric series](/blog/geometric-series). For more derivatives practice, read the guide to [derivatives of exponential and logarithmic functions](/blog/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of a^x?

The derivative of $a^x$ is $a^x\ln a$. For example, $\frac{d}{dx}2^x = 2^x\ln 2$. When $a = e$, $\ln e = 1$ and the formula reduces to $\frac{d}{dx}e^x = e^x$.

### What is the integral of a^x?

$\int a^x\,dx = \frac{a^x}{\ln a} + C$ for $a > 0$, $a \ne 1$. For example, $\int 3^x\,dx = \frac{3^x}{\ln 3} + C$.

### What is the difference between exponential growth and decay?

If the base $b > 1$, the function grows as $x$ increases (exponential growth). If $0 < b < 1$, it shrinks toward $0$ (exponential decay). In the form $Ae^{kt}$, growth means $k > 0$ and decay means $k < 0$.

### Why can't you use the power rule on 2^x?

The power rule is for a variable base with a constant exponent, like $x^2$. In $2^x$ the base is constant and the exponent varies, so you need the exponential rule $\frac{d}{dx}2^x = 2^x\ln 2$.