In this article, we will show you how to derive the natural exponential function, also known as the e-function. You need this derivation in several areas, such as the extreme points or turning points.

If you would like to see the properties of the e-function again, read the chapter «Exponential function». There you will find everything you need to know about this function.

## General information on the derivation of the e-function

It is already known that the e-function arises from the exponential function. Therefore we first look at the general exponential function in its pure form f(x)=ax.

f(x)=ax→derive f'(x)=ln(a) ax

### Derive pure exponential function

You already know what comes out when you do the **exponential function**** derive**. Let’s recap the whole thing mathematically.

the **derivation** f'(x) **the general exponential function **f(x)=ax is:

f'(x)=ln(a)*ax

If you want to learn how **Derivative f'(x) of the exponential function** comes about, you can look at the next in-depth section.

You can derive the derivative f'(x) using the differential quotient. So that you are well prepared for this, you should know the contents of the articles differential quotient and powers.

The derivative f'(x) is defined using the differential quotient as follows.

f'(x)=limh→0f(x+h)-f(x)h

If you now use the general exponential function, you get the following expression.

f'(x)=limh→0ax+h-axh

At this point you can apply the calculation rules for powers.

As a reminder: xa+b=xa xb

This results in the following:

f'(x)=limh→0ax ah-axh

Now you can factor out ax and apply the calculation rules for limit values.

f'(x)=limh→0ax ah-axh=limh→0ax (ah-1)h=ax limh→0ah-1h

Now you would have to form the limit value for the expression limh→0ah-1h, which corresponds to a constant. However, since it would lead too far at this point, this value is given to you.

limh→0ah-1h=ln(a)

This gives you the following derivation f'(x) for the general exponential function:

f'(x)=ax limh→0ah-1h=ax ln(a)

### Derive pure e-function

the **e function** is a special exponential function, where the base a corresponds to Euler’s number e. Let us now formulate the derivative f'(x) of the e-function.

You can derive the pure e-function f(x)=ex as many times as you want, it will never change. As a small mnemonic you can remember: «Stay as you are – like the e-function when deriving!».

If you want to know why the **e function derived** is the e-function again, you can look at the next in-depth section.

Here you have to consider the derivative f'(x) of the general exponential function.

f'(x)=ln(a)*ax

You now use Euler’s number e for the base a and get the following expression.

f'(x)=ln(e)·ex

Then you have to determine the expression ln(e). You already know this one.

ln(e)=1

This results in the following derivation f'(x) for the e-function:

f'(x)=1*ex=ex

Often you don’t have the pure version of the e-function in tasks, but with different parameters. You can find out how to derive these in the next section.

## Derivatives of the extended e-function

More interesting is the **Derivation of the extended e-function** With **parameters**. You mainly need these if you are supposed to calculate extreme points and turning points.

In memory of:

- Extended e-function: f(x)=b ecx
- The parameters b and c must never be 0, otherwise there is no longer an e function.
- If both parameters are 1, the e-function is again in its pure version f(x)=ex.

### Derive e-function with prefactor

First look at those **e function **with a prefactor b while c=1.

f(x)=b ex

You have to use the factor rule.

Here is the factor rule as a reminder: f(x)=a g(x)→derive f'(x)=a g'(x)

Since you know that the **Derivation of the e-function** is the e-function, you get the following derivation of the function f(x)=b ex.

f'(x)=b·ex

So you can also derive the e-function with a prefactor f'(x)=b·ex as often as you want, it will never change.

Let’s capture this insight in a definition.

the** derivation **f'(x) **the e function** with a prefactor f(x)=b ex is:

f'(x)=b·ex

Apply the learned derivation immediately the e-function with prefactor on this exercise:

**Task 1**

Derive the function f(x) with f(x)=9·ex.

**solution**

Since an e-function does not change with a prefactor, you get the following derivative f'(x).

f'(x)=9*ex

### Derive e-function with chain rule

Now you can **Derivative f'(x) for the entire extended e-function** form f(x)=b ecx. For this you need the chain rule and the factor rule.

As a reminder, the chain rule is: f(x)=g(h(x))→derive f'(x)=g'(h(x)) h'(x)

To apply the chain rule, you must first define the outer function g(x) and the inner function h(x).

g(x)=eh(x)=ecxh(x)=cx

You then need the derivation of each of these functions. Since the e-function again results in the e-function, the following derivations are formed.

g'(x)=ecxh'(x)=c

Now you can apply the last steps of the chain rule. In addition, you have to consider the pre-factor b with the factor rule to get the derivative f'(x) for the entire extended e-function. This results in the following overall derivative f'(x) for the extended e-function.

f'(x)=b g'(h(x)) h'(x)=b g'(cx) c=b ecx c=bc ecx

Whenever the exponent is not just «x», you have to apply the chain rule.

Let’s keep it all in one definition.

the **derivation** f'(x) **the extended e-function** f(x)=b ecx is:

f'(x)=bc*ecx

Here, too, first apply your newly acquired knowledge **Derivation of the extended e-function **on an example.

**exercise 2**

Derive the function f(x) with f(x)=3 e14x.

**solution **

First identify the parameter c.

c=14

Next, you can directly apply the formula for the derivative of the extended e-function. You then get the following derivation f'(x) of the function f(x)=3 e14x.

f'(x)=3*14*e14x=42e14x

### Derive e-function with product rule – exercises

There are often functions in which not only an e-function occurs, but this is multiplied by another function. um to be prepared for such a task, slook at the next exercise.

**task 3**

Derive the function f(x) with f(x)=e4x x2.

**solution**

First of all, you need the product rule.

Product rule: f(x)=g(x) h(x)→derive f'(x)=g'(x) h(x)+g(x) h'(x)

To do this, we identify the functions g(x) and h(x).

g(x)=e4xh(x)=x2

The following individual derivations result.

g'(x)=4 e4xh'(x)=2x

This results in the following overall derivation f'(x).

f'(x)=4 e4x x2+e4x 2x=2 e4x (2×2+x)

## Derive e-function – the most important thing

- The derivative f'(x) of the general exponential function f(x)=ax reads: f'(x)=ln(a)·ax
- The derivative f'(x) of the pure e-function f(x)=ex is: f'(x)
- A helpful mnemonic: «Stay as you are – like the e-function when deriving!»

- The derivative f'(x) of the e-function with a prefactor f(x)=b·ex is: f'(x)=b·ex
- The derivative f'(x) of the extended e-function f(x)=b ecx is: f'(x)=bc ecx
- Whenever the exponent is not just «x», you have to apply the chain rule.