]> Haese & Harris - Chapter 23 Notes

23: Derivatives of Exponential and Logarithmic Functions

A: Derivatives of Exponential Functions

First of all, a definition: an exponential function is of the form $f (x) = n^{x}$ where $n \in ℝ^{+}$ and $n \neq 1$ .

You should know that an exponential function passes through the points $(0, 1)$ and $(1, n)$ , and that it will have a horizontal asymptote on one side.

While there are an infinite number of possible bases, there is only one of consequence—the natural number (Euler's number) e. The natural number is defined as $lim_{n \to ∞} {(1 + \frac{1}{n})}^{n}$ .

Now—how about the derivative of $y = e^{x}$ ?

$y^{'} = lim_{h \to 0} \frac{e^{x + h} - e^{x}}{h} = lim_{h \to 0} \frac{e^{x} e^{h} - e^{x}}{h} = lim_{h \to 0} \frac{e^{x} (e^{h} - 1)}{h}$ .

Now, since h is what's changing, the $e^{x}$ part is like a constant—factor it out.

$e^{x} lim_{h \to 0} \frac{e^{h} - 1}{h}$

So—all we need to know now is what $lim_{h \to 0} \frac{e^{h} - 1}{h}$ comes out to…why don't we graph it?

An Image

Zoom in all you like—that limit comes out to 1.

Okay, okay—that's not a proof. It will do, though.

So finally, the derivative comes out to $y^{'} = e^{x}$ .

Wow! A function which is its own derivative! Fascinating…

B: Using Natural Logarithms

First, a definition: a logarithm (base n) is the inverse function of the exponential function (base n). If $f (x) = n^{x}$ , then $f^{-1} (x) = {log}_{n} (x)$ . If $y = n^{x}$ , then $x = {log}_{n} (y)$ .

Since they are inverse functions, they cancel each other out when composed: $n^{{log}_{n} (x)} = x$ and ${log}_{n} (n^{x}) = x$ .

Again, there are an infinite number of bases, but there's only one that matters: the natural! The natural logarithm is $y = {log}_{e} (x) = ln (x)$ . The latter notation is common in high schools; for many university-level people, writing $y = log (x)$ assumes a base of e.

Yes, I know—that's confusing to you! You've always been taught that a missing log base means base 10 (common log).

Unlearn that.

Besides—common logs are just so…common…

Natural logs are much more…natural!

Now—to finish section A.

That's right—we never finished section A. We found the derivative of one exponential function, but not all of them—time to finish that up. The catch is to write a generic exponential function in terms of e…how about $y = e^{ln (n^{x})}$ ?

First, work some mathemagic on that: $y = e^{ln (n^{x})} ⇒ y = e^{x ln (n)}$ remember that property of logarithms?). Also note that $ln (n)$ is a constant, just like 2 or 17 or e.

Okay—the derivative will require a chain rule. $y = e^{x ln (n)} ⇒ y^{'} = e^{x ln (n)} \cdot ln (n) = n^{x} \cdot ln (n)$ .

C: Derivatives of Logarithmic Functions

Time to find the derivative of a logarithmic function! Again, let's write it as an exponential. $y = ln (x) ⇒ x = e^{y}$ . Now, take the derivative: $1 = e^{y} \cdot \frac{d y}{d x}$ . Note that $e^{y} = x$ , so what we've got is $1 = x \cdot \frac{d y}{d x} ⇒ \frac{1}{x} = \frac{d y}{d x}$ .

What about the other bases, you ask? For those, you must remember the change of base formula: ${log}_{n} (x) = \frac{ln (x)}{ln (n)}$ . Again, remember that $ln (n)$ is just a constant. You should be able to take it from there!

D: Applications

How about everything we've done to date, just with exponentials and logs thrown in?

Page last validated 2010-08-15