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, as we will prove next, it is a well-known property of e that:

Substituting this into the previous equation gives:

This proves the derivative of the exponential function.

Proof of the limit property of e

We aim to prove that the following limit, L, is equal to 1. If we can prove this, then it proves the derivative of the exponential function (based on the previous section):

We can express 1/L as:

We will use a variable substitution x. We can then express e^h and h in terms of x:

We know that, as h tends to 0, e^h tends to 1. From the first equation above, that means that, as h tends to 0, x also tends to 0. So we can write the limit for 1/L in terms of x rather than h:

We can use the power rule of logarithms to simplify this:

Since the ln function is continuous, we can move the limit inside the function:

You might recognise this limit as being the definition of e:

This means that 1/L is 1, so of course L is 1:

This proves that the original limit is equal to 1.

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 from 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.