A.2 Derivative of Exponential Functions

On this screen we're going to use Desmos to examine the derivative of exponential functions like $2 𝑥,$ $3.25 𝑥,$ and such.

Exploration of the derivative of $𝑎 𝑥$

Let's use Desmos to examine the derivative of $𝑎 𝑥$ for various values of $𝑎$ .

Exploration 1

[End Exploration 1]

Let's see how the discovery we made in Exploration 1 follows from the definition of the derivative as applied to the exponential function $𝑓 (𝑥) = 𝑎 𝑥 .$ Recall the definition of the derivative:

𝑓' (𝑥) = l i m ℎ \to 0 𝑓 (𝑥 + ℎ) - 𝑓 (𝑥) ℎ

We are considering

𝑓 (𝑥) = 𝑎 𝑥,

𝑓 (𝑎 + ℎ) = 𝑥 𝑥 + ℎ .

Making these substitutions in the derivative definition then gives us

𝑓' (𝑥) = 𝑑 𝑑 𝑥 𝑎 𝑥 = l i m ℎ \to 0 𝑎 𝑥 + ℎ - 𝑎 𝑥 ℎ = l i m ℎ \to 0 𝑎 𝑥 𝑎 ℎ - 𝑎 𝑥 ℎ = l i m ℎ \to 0 𝑎 𝑥 (𝑎 ℎ - 1) ℎ [𝑎 𝑥 i s u n a f f e c t e d b y t h e l i m i t] = 𝑎 𝑥 l i m ℎ \to 0 𝑎 ℎ - 1 ℎ

The preceding equation, combined with the discussion in Exploration 1, provides one way to define the number e:

Definition of e

e is defined to be the number such that

l i m ℎ \to 0 𝑒 ℎ - 1 ℎ = 1

To five digits, this number is $𝑒 = 2.71828 .$

With this definition of e in place, we have the key result we discovered in Exploration 1:

Derivative of $𝑒 𝑥$

𝑑 𝑑 𝑥 𝑒 𝑥 = 𝑒 𝑥

We all love this particular derivative, since it's so easy to remember!

One reason e appears so often in describing physical phenomena

More importantly, this result is the first indication of why the number e appears so often when we describe physical phenomena: The rate of change of the function $𝑓 (𝑥) = 𝐶 𝑒 𝑥$ is proportional to the value of the function itself. For instance, the way that bacterial growth occurs is that each bacterial cell subdivides so 1 becomes 2, and 2 become 4, and so on. That means that the rate at which they multiply at a given moment, $𝑑 𝑁 𝑑 𝑡,$ is directly proportional to how many are present at that time, $𝑁 (𝑡)$ : they grow at twice the rate when there are 2 present as when there are 1, and four times the rate when there are 4 present instead of 1, and sixteen times the rate when there are 16 present than when there is 1. That is, $𝑑 𝑁 𝑑 𝑡 = 𝑘 𝑁 (𝑡),$ where the constant k depends on the particular type of bacteria and how quickly they subdivide. This direct proportionality between $𝑑 𝑁 𝑑 𝑡$ and $𝑁 (𝑡)$ leads straight to the bacterial growth equation $𝑁 (𝑡) = 𝑁 0 𝑒 𝑘 𝑡 .$ We'll explore this and related ideas in much more depth later. For now, you might choose to marvel at how this number e has the remarkable property that $𝑑 𝑑 𝑥 𝐶 𝑒 𝑥 = 𝐶 𝑒 𝑥 .$

Derivative of $𝑎 𝑥$ for other values of $𝑎$

For completeness, here is a result that we'll be able to prove easily in a few screens. For now, we state the result and provide a more cumbersome proof in the Show/Hide box immediately below.

Derivative of $𝑎 𝑥$

For any value of

𝑎 > 0 :

𝑑 𝑑 𝑥 𝑎 𝑥 = 𝑎 𝑥 \cdot l n 𝑎

This result matches what we saw in Exploration 1 above: the function $𝑓 (𝑥) = 𝑎 𝑥$ and its derivative have the same-shaped curve because they differ only by the (constant) factor $l n 𝑎 .$ And since $l n 𝑒 = 1,$ when $𝑎 = 𝑒$ the function $𝑓 (𝑥) = 𝑒 𝑥$ and its derivative are identical.

The Desmos calculator below let's you examine the function $𝑓 (𝑥) = 𝑎 𝑥$ (solid curve) and its derivative $𝑓' (𝑥) = 𝑎 𝑥 \cdot l n 𝑎$ (dashed curve) on the same plot, as we did in Exploration 1. Now there is a slider beneath the graph that lets you see what happens for various values of a.

Practice Problems for the derivative of exponential functions

What questions or thoughts do you have about the material on this screen, or any other Calculus-related item? How are these problems: easy? challenging? Please let the Community know on our Forum!

The Upshot

The easiest derivative of all to remember: $𝑑 𝑑 𝑥 𝑒 𝑥 = 𝑒 𝑥 .$
Almost as easy: $𝑑 𝑑 𝑥 𝑎 𝑥 = 𝑎 𝑥 \cdot l n 𝑎$

On the next screen , we'll add the derivatives of two trig functions to our repertoire:

𝑑 𝑑 𝑥 s i n 𝑥

and

𝑑 𝑑 𝑥 c o s 𝑥 .

A.2 Derivative of Exponential Functions

Exploration of the derivative of 𝑎𝑥

Derivative of 𝑎𝑥 for other values of 𝑎

Practice Problems for the derivative of exponential functions

The Upshot

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Exploration of the derivative of $𝑎 𝑥$

Derivative of $𝑎 𝑥$ for other values of $𝑎$