Derivative of e^x: Proof and Examples

The exponential function is one of the most important functions in calculus. In this page we'll deduce the expression for the derivative of e^x and apply it to calculate the derivative of other exponential functions.

Our first contact with number e and the exponential function was on the page about continuous compound interest and number e. In that page, we gave an intuitive definition of number e, and also an intuitive definition of the exponential function.

We also deduced an alternative expression for the exponential function. The new expression for the exponential function was a series, that is, an infinite sum.

You may ask, the limit definition is much more compact and simple than that ugly infinite sum, why bother?

It turn out that the easiest way to deduce a rule for taking the derivative of e^x is using that infinite series representation. Why is that? The series expression for e^x looks just like a polynomial.

We can generalize the idea of a polynomial by allowing an infinite number of terms, just like in the expression for the exponential function. An infinite polynomial is called a power series.  

The neat thing about a power series is that to calculate its derivative you proceed just like you would with a polynomial. That is, you take the derivative term by term. Let's do that with the exponential function.

The Derivative of e^x

We consider the series expression for the exponential function

We can calculate the derivative of the left side by applying the rule for the derivative of a sum. That is, the derivative of a sum equals the sum of the derivatives of each term

We know the derivatives of each of those terms

I added an extra term to make the pattern clear. Now, there are some numbers that cancel out

We obtained a surprising result. The expression for the derivative is the same as the one for the original function. That is

The derivative of e^x is e^x. This is one of the properties that makes the exponential function really important.

Now you can forget for a while the series expression for the exponential. We only needed it here to prove the result above. We can now apply that to calculate the derivative of other functions involving the exponential.

Example 1: f(x) = e^ax

Let's calculate the derivative of the function

At first sight it may not be obvious, but this is a composite function. This means we need to apply the chain rule. The outer function is the exponential. Its derivative equals itslef. The inner function is ax:

That was simple. It may take a few more examples to get used to the fact that the derivative of an exponential is the same exponential.

Example 2: f(x) = e^x2

Let's consider now another composite function

To calculate its derivative we apply again the chain rule. As the outer function is the exponential, its derivative equals itself

Example 3: f(x) = e^x(1-x²)

Now, this one looks more complicated

Here we have a product, so we must use the product rule. To do that, we identify the two factors

And we apply the product rule

And now we factor e^x to obtain the final answer

Example 4: f(x) = e^cos(x)sin(x)

Let's consider the following function

This one requires more attention because we need to apply both the product rule and chain rule. Let's see what I mean. First, we apply the product rule

Now, to calculate u', we need to apply the chain rule

We plug this into the product rule

Example 5: Exponential With Other Base, f(x)=a^x

Now let's consider an exponential with a base that is not e.

How do we calculate the derivative of this function? We use a trick that is regularly used when dealing with logarithms. We can write this function as

You can check that this equality is true by using the definition of logarithm. Now we take advantage of the property of logarithms that allows us to take exponents out of the log sign

Now this is an exponential function with base e, whose derivative we know how to calculate.

But using an equation a few lines above, we can write this as

This one shows one of the reasons the natural choice for the base of an exponential function is number e. For any other base, you get that ln(a) littering the expression of its derivative.

Example 6: f(x) = a^x2

Let's consider

Here we need to apply the chain rule. The outer function is the exponential, so we know how to calculate its derivative from the previous example

That's is. Your next step may be to learn about the derivative of ln(x). If you have any doubt or want to discuss a problem of your own, just leave me a comment below.