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A.2 Derivative of Exponential Functions

On this screen we’re going to examine the derivative of exponential functions like $2^x,$ $3.25^x,$ and such. The following Exploration examines one very special such exponential function.

EXPLORATION 1: Derivative of $a^x$ and discovery of . . .

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 from the preceding Chapter, “The Derivative”:
\[\textbf{Definition of the Derivative:}\qquad f'(x) = \lim_{h \to 0}\frac{f(x+h) – f(x)}{h} \] Since we have $f(x) = a^x,$ $f(a+h) = x^{x+h}.$ Making these substitutions into the definition of the derivative then gives us
\begin{align*}
f'(x) = \dfrac{d}{dx}a^x &= \lim_{h \to 0}\frac{a^{x+h} – a^x}{h} \\[8px] &= \lim_{h \to 0}\frac{a^x a^h – a^x}{h} \\[8px] &= \lim_{h \to 0}\frac{a^x \left(a^h – 1 \right)}{h} \quad \left[a^x\text{ is unaffected by the limit} \right] \\[8px] &= a^x \lim_{h \to 0}\frac{a^h – 1}{h}
\end{align*}
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
\[\lim_{h \to 0}\frac{e^h – 1}{h} = 1\] To five digits, this number is $e = 2.71828.$

 

Show/Hide Desmos calculator to explore limit for different values of a

The calculator below plots $\dfrac{a^h – 1}{h}$ versus h, so you can see visually what the values are as $\displaystyle{\lim_{h \to 0}}.$ Use the slider beneath the calculator to change the value of a: you’ll find that for $a = e \approx 2.72,$ the limit equals 1.

Graph of $\displaystyle{\lim_{h \to 0}\frac{a^h – 1}{h}}$ versus h
Use the slider to change the value of a:

Currently $a = 1.95.$

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

Derivative of $e^x$
\[\dfrac{d}{dx}e^x = e^x\]

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 very often when we describe physical phenomena: the rate at which the function $f(x) = Ce^x$ changes for a given value of x is proportional to the value of the function itself. Bacterial growth is one such situation: since each bacterial cell subdivides so 1 becomes 2, and 2 become 4, and so on, the rate at which they multiply at a given moment, $\dfrac{dN}{dt},$ depends on how many are present at that time. $N(t).$ That is, $\dfrac{dN}{dt} = kN(t),$ where the constant k depends on the particular type of bacteria and how quickly they subdivide. This direct proportionality between $\dfrac{dN}{dt}$ and $N(t)$ leads straight to the bacterial growth equation $N(t) = N_0 e^{kt}.$ 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 $\dfrac{d}{dx}Ce^x = Ce^x.$

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 $a \gt 0:$
\[\dfrac{d}{dx}a^x = a^x \cdot \ln a\]

 

Show/Hide development of the preceding result

We’ll be able to develop this result quite easily once we have the “Chain Rule” tool in our repertoire a few screens from now. In the meantime:

The first few lines duplicate what we did above, since we once again start with the definition of the derivative applied to the function $f(x) = a^x.$
\begin{align*}
f'(x) &= \lim_{h \to 0}\frac{a^{x+h} – a^x}{h} \\[8px] &= \lim_{h \to 0}\frac{a^x a^h – a^x}{h} \\[8px] &= \lim_{h \to 0}\frac{a^x \left(a^h – 1 \right)}{h} \quad \left[a^x\text{ is unaffected by the limit} \right] \\[8px] &= a^x \, \lim_{h \to 0}\frac{a^h – 1}{h}
\end{align*}
To proceed, recall that $a = e^{\ln a}.$
\begin{align*}
&= a^x \, \lim_{h \to 0}\frac{\left(e^{\ln a} \right)^h – 1}{h}\phantom{\quad \left[a^x\text{ is unaffected by the limit} \right]} \\[8px] &= a^x \, \lim_{h \to 0}\frac{e^{h\ln a} – 1}{h} \\[8px] \end{align*}
To proceed further, we make the substitution $u = h\ln a \implies h = \dfrac{u}{\ln a}.$
Note that $h \to 0 \implies u \to 0.$
Let’s make the substitution $h = \dfrac{u}{\ln a}:$
\begin{align*}
&= a^x \, \lim_{u \to 0}\frac{e^{u}\, – 1}{\frac{u}{\ln a}} \phantom{\quad \left[a^x\text{ is unaffected by the limit} \right]} \\[8px] &= a^x \, \overbrace{\lim_{u \to 0}\frac{e^{u}\, – 1}{u}}^{1} \, \ln a \quad \left[ \text{Recall from the definition of }e: \lim_{u \to 0}\frac{e^u\, – 1}{u} = 1 \right] \\[8px] f'(x) = \dfrac{d}{dx}a^x &= a^x \, \ln a \quad \cmark
\end{align*}

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

The Desmos calculator below let’s you examine the function $a^x$ and its derivative 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.

Graph of $f(x) = a^x$ and $f'(x) = a^x \cdot \ln a$ versus x
Use the slider to change the value of a:

Currently $f(x) = 1.95^x$

Practice Problems

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The Upshot

  1. The easiest derivative of all to remember: $\dfrac{d}{dx}e^x = e^x.$
  2. Almost as easy: $\dfrac{d}{dx}a^x = a^x \cdot \ln a$


On the next screen, we’ll add the derivatives of two trig functions to our repertoire: $\dfrac{d}{dx}\sin x$ and $\dfrac{d}{dx}\cos x.$