Derivative of the exponential function

By Martin McBride, 2026-01-07

Tags: exponential derivative
Categories: exponentials differentiation

The exponential function, based on Euler's number, e, has the property 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 change of x 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 e^x, this gives us:

From the rules of exponents we know that e^{x + h} is e^xe^h so we can extract the common factor of e^x:

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

We are going to do a simple variable substitution, but since it will also alter the limit condition, we need to take a bit of care. We are going to substitute h for 1/n. These two expressions are identical, provided h = 1/n:

That is a simple fact of algebra. For any n we choose, we can find the value h = 1/n, and if we use those two values then the two sides of the equation will be equal.

In particular, if we choose a very large value of n, then 1/h will have a very small value, but the two sides of the equation will be equal. Now we know that, as n gets larger, the expression approaches a limiting value of e. In fact, if we take a large value of n, and then keep doubling it, over and over, then the expression will get ever closer to e.

But as we keep doubling n, we must also keep halving h. So h must get smaller and smaller. But the h expression remains equal to the n expression, so it too must move ever closer to e. We can equate the limit of the h expression (as h tends to 0 rather than infinity) to e:

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

Again from the rules of exponents we know that (e^a)^b is e^ab 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 e^x 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 π.

The value e, like π, is what it is - a strange irrational number that arises out of a simple situation. In fact, e and π are closely related. π arises of a simple unit second order differential equation see Pi isn't about circles, in a similar way that e arises out of a simple first order differential equation. Also, e and π are related by Euler's identity.