# Derivative of the exponential function

By Martin McBride, 2022-08-13
Tags: exponential derivative
Categories: exponentials differentiation

In the article what is e?, we stated that:

This is an important result. It tells us that the exponential function is its own derivative. This means that we can use it to solve equations where the rate of x change is proportional to the value of x.

In this article, we will prove this result, and look at some of the consequences.

## Derivative of the exponential function

To prove the result we will start from first principles. The derivative of a function y = f(x) is given by:

In the case of the exponential function ex, this gives us:

From the rules of exponents we know that ex + h is exeh so we can extract the common factor of ex:

Now let's look at the definition of e we developed in the formula for e article:

We can substitute the value 1/h = n is this formula:

Notice that we have changed the limit. The equation in n applies as n tends to infinity. And n tends to infinity, h = 1/n tends to zero.

We can substitute this value for e into the main equation:

Again from the rules of exoponents we know that (ea)b is eab so we can simplify this term (the power of 1/h and the power of h cancel each other out):

This further simplifies to:

The limit of h/h as h tends to zero is 1, so we get the final result:

## Why e?

You might still be wondering why the magic value of e is involved in this equation. Well if we go back to this equation:

We see that the crucial fact is that the limit term is 1:

We won't prove it here, but for this limit to be true essentially requires ex to have a slope of 1 at x = 0.

These graphs show exponential functions with bases of 2, e and 5, with a dotted black line of slope 1:

Base e has a slope of 1 when x is zero. Base 2 has a slope of less than 1, base 5 has a slope of greater than 1.

## A new definition of e

We can now provide an alternative definition of e:

e is the base value for which the function a to the power x has a slope of 1 when x is zero.

Given this new definition, it is a lot easier to understand why e has such special properties.

But why is this definition valid? If you recall the article formula for e, our starting point was a bank account that paid 100% per year. So £1 turned into £2 in the first year, giving a slope of £1/year.

We deliberately set the slope to 1, and the value of e arose out of that initial condition.

If we draw an exponential function with a slope of 1 at x = 0, that function will have a base of e. This is similar to saying that if you draw a circle of radius 1 it will have an area of pi.

The value e, like pi, is what it is - a strange irrational number that arises out of a simple situation.

## Integration

Since the derivative of ex is ex, it follows that the integral is also ex.

More precisely, the indefinite integral of ex is:

Where C is the constant of integration.

Since the function tends to zero as x tends to minus infinity, we have the following definite integral:

In other words, the area under the curve from a, stretching back to minus infinity, is e to the power a*. This is shown graphically here:

The area A under the curve is equal to the value of the curve