The function y=e^{x} is called the exponential function. The derivative of the exponential function e^{x} is equal to e^{x}. This also means that the integral of e^{x} is e^{x}. Compound exponential functions can be differentiated with the chain rule.

Here, we will learn why the derivative of e^{x} is e^{x}. Also, we will learn how to find the derivative of compound exponential functions with some examples.

##### CALCULUS

**Relevant for**…

Learning about derivatives and integrals of exponential functions.

##### CALCULUS

**Relevant for**…

Learning about derivatives and integrals of exponential functions.

## Why e^{x} is the derivative of e^{x}?

The derivative of the exponential function e^{x} is e^{x} because the slope of the curve e^{x} is equal to e^{x}.

We can visualize this idea by considering the graph of the function $latex y=2^x$ shown below:

The points $latex P(x,~y)$ and $latex Q(x+\delta x,~y+\delta y)$ are two close points that lie on the curve $latex y=2^x$. Then, we have:

$latex y+\delta y=2^{x+\delta x}$

$latex \delta y=2^{x+\delta x}-y$

$latex =2^{x+\delta x}-2^x$

$latex \delta y=2^x(2^{\delta x}-1)$

If we divide both sides of this equation by $latex \delta x$, we have:

$$\frac{\delta y}{\delta x}=2^x\left(\frac{2^{\delta x}-1}{\delta x}\right)$$

Considering that $latex \frac{dy}{dx}=\lim_{\delta x \to 0}\left[\frac{\delta y}{\delta x}\right]$, we have:

$$\frac{dy}{dx}=\lim_{\delta x \to 0}\left[ 2^x\left(\frac{2^{\delta x}-1}{\delta x}\right)\right]$$

$$=2^x \times \lim_{\delta x \to 0}\left(\frac{2^{\delta x}-1}{\delta x}\right)$$

Taking very small values of $latex \delta x$ and using a calculator, we have:

$$\lim_{\delta x \to 0}\left(\frac{2^{\delta x}-1}{\delta x}\right)\approx 0.693$$

This means that the derivative of $latex 2^x$ is:

$$\dfrac{d}{dx}(2^x)\approx 0.693 \times 2^x$$

If we do the same for $latex 3^x$, we have:

$$\frac{d}{dx}(3^x)=3^x \times \lim_{\delta x \to 0}\left(\frac{3^{\delta x}-1}{\delta x}\right)$$

Taking very small values of $latex \delta x$ and using a calculator, we have:

$$\lim_{\delta x \to 0}\left(\frac{3^{\delta x}-1}{\delta x}\right)\approx 1.099$$

$$\dfrac{d}{dx}(3^x)\approx 1.099 \times 3^x$$

We can observe that for $latex y=2^x$, the slope of the curve ($latex \frac{dy}{dx}$) is less than $latex y=2^x$ and in the case of $latex y=3^x$, the slope is greater than $latex y=3^x$.

There exists an exponential function $latex y=a^x$ in which its slope is equal to $latex y=a^x$. This value is $latex a=2.71828$ and is denoted by the symbol $latex e$.

## Derivatives and integrals of exponential functions – Examples with answers

**EXAMPLE **1

**EXAMPLE**

Find the derivative of the function $latex y=e^{3x}$.

##### Solution

We know that the derivative of the function $latex y=e^x$ is $latex \frac{dy}{dx}=e^x$.

Then, we use the chain rule to derive:

$$\frac{dy}{dx}=e^{2x}(2x)^{\prime}$$

$$\frac{dy}{dx}=2 e^{2x}$$

**EXAMPLE **2

**EXAMPLE**What is the derivative of the function $latex y=5e^{\frac{1}{x}}$?

##### Solution

Using the chain rule, we have:

$$\frac{dy}{dx}=5e^{\frac{1}{x}}\left(\frac{1}{x}\right)^{\prime}$$

$$=5e^{\frac{1}{x}}\left(-\frac{1}{x^2}\right)$$

$$\frac{dy}{dx}=-\frac{5}{x^2}e^{\frac{1}{x}}$$

**EXAMPLE **3

**EXAMPLE**Find the derivative of the following function:

$$y=\frac{2}{3+e^{3x}}$$

##### Solution

We can write the function as follows:

$latex y=2(3+e^{3x})^{-1}$

Now, we use the chain rule:

$$\dfrac{dy}{dx}=-2(3+e^{3x})^{-2}(3+e^{3x})^{\prime}$$

$$=-2(3+e^{3x})^{-2}(3e^{3x})$$

$$\dfrac{dy}{dx}=\frac{-6e^{3x}}{(3+e^{3x})^2}$$

**EXAMPLE **4

**EXAMPLE**Solve the following integral:

$$\int 2e^{-x} dx$$

##### Solution

To solve this integral, we observe that the derivative of $latex e^{-x}$ is $latex -e^{-x}$. Then, we have:

$$\int 2e^{-x} dx =-2e^{-x}+c$$

**EXAMPLE **5

**EXAMPLE**Find the result of the following integral:

$$\int (1-e^{-3x})^2 dx$$

##### Solution

We can start by expanding the parenthesis expression to get:

$$\int (1-e^{-3x})^2 dx = \int (1-2e^{-3x}+e^{-6x}) dx$$

Then, we have:

$$\int (1-e^{-3x})^2 dx = x+\frac{2}{3}e^{-3x}-\frac{e^{-6x}}{6}+c$$

**EXAMPLE **6

**EXAMPLE**Find the derivative of the following function:

$latex y=x^2e^x$

##### Solution

In this case, we can use the rule of the product of derivatives and we have:

$$\dfrac{dy}{dx}=x^2(e^x)^{\prime}+e^x(x^2)^{\prime}$$

$$=x^2e^x+2xe^x$$

$$\dfrac{dy}{dx}=xe^x(x+2)$$

## Derivatives and integrals of exponential functions – Practice problems

## See also

Interested in learning more about derivatives? You can take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**