How can the first principles approach be used to differentiate a^x?

  • Thread starter mtanti
  • Start date
In summary: So differentiating from first principles is the wrong way to look at it; rather, it's using the definition of e, which is the limit I've mentioned, to differentiate the exponential function from first principles. Anyway, Radou's done a good job of summarising the whole thing.
  • #1
mtanti
172
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I always wondered how you can differentiate a^x from first principles with the limit as dx approaches zero but I never managed to simplify it far enough to separate dx on a different term. Can anyone help?
 
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  • #2
I guess it should look like that:

[tex]
\begin{equation*}
\begin{split}
&f(x) = a^x \\
f'(x) &= lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=lim_{h\rightarrow 0}\frac{a^{(x+h)}-a^x}{h} = \\
&= lim_{h\rightarrow 0}\frac{a^xa^h-a^x}{h} = lim_{h\rightarrow 0}\frac{a^x(a^h-1)}{h} = \\
&= a^x lim_{h\rightarrow 0}\frac{a^h-1}{h} = a^x lna \\
\end{split}
\end{equation*}
[/tex]
 
Last edited:
  • #3
You can always differentiate it using all of the tools you know, and then translate those tools into epsilon-deltas.

radou: try something like:

\begin{equation*}
\begin{split}
f(x) &= a^x \\
f'(x) &= ... \\
&= ... \\
&= ...
\end{split}
\end{equation*}
 
  • #4
Thanks. :smile:
 
  • #5
radou, you might want to break that process into separate lines... I can't see half of it! :)
 
  • #6
Of course, the hard part is showing that
[tex]\lim_{h\rightarrow 0}\frac{a^h-1}{h}= ln(a)[/tex]

Many modern texts start by defining
[tex]ln x= \int_1^x\frac{1}{t}dt[/tex]
showing that this has all the properties of a natural logarithm, has an inverse, and then defining ex as its inverse.
 
  • #7
HallsofIvy said:
Of course, the hard part is showing that
[tex]\lim_{h\rightarrow 0}\frac{a^h-1}{h}= ln(a)...[/tex]

Exactly. That's why I didn't prove it. :biggrin:
 
  • #8
Hey but that's not fair, any up to standard student can get there on his/her own! How do you actually solve the hard part? I think it's interesting for all those starting calculus...

Tell me this at least... Can you diffentiate *every* equation from first principles? Even implicite ones?
 
  • #9
mtanti said:
Hey but that's not fair, any up to standard student can get there on his/her own! How do you actually solve the hard part? I think it's interesting for all those starting calculus...

Tell me this at least... Can you diffentiate *every* equation from first principles? Even implicite ones?
The simplest way to prove this rigourously is by introducing the ugly-looking function Exp(x):
[tex]Exp(x)=1+\sum_{n=1}^{\infty}\frac{x^{n}}{n!}[/tex]
where integral powers of numbers have been defined inductively.
Exp(x) can be shown to have all the properties we would like an exponential function to have, including an inverse we call Log(x).
Furthermore, we can differentiate Exp(x) termwise, yielding..Exp(x) itself.

We therefore DEFINE [itex]a^{x}=Exp(x*Log(a))[/itex]
and we may differentiate this by the use of the chain rule.
 
  • #10
If we differentiate log x (to any base) from first principles, after a few lines of algebra we find it to be (1/x) times the log of the limit as h approaches zero of (1 + h)^(1/h). But this limit is the familiar definition of e. Job done. Using this result its then easy to find (using chain rule or whatever) the derivative of the inverse function, i.e. the exponential function.
I'd say this counts as differentiating the exponential function from first principles. The limit Radou uses, is in my opinion, less well known that the limit I've referred to above.
 
  • #11
Philip Wood said:
If we differentiate log x (to any base) from first principles, after a few lines of algebra we find it to be (1/x) times the log of the limit as h approaches zero of (1 + h)^(1/h). But this limit is the familiar definition of e. Job done. Using this result its then easy to find (using chain rule or whatever) the derivative of the inverse function, i.e. the exponential function.
I'd say this counts as differentiating the exponential function from first principles. The limit Radou uses, is in my opinion, less well known that the limit I've referred to above.

Hi Philip. I'm pretty sure that lim h->0 (1 + h)^(1/h) = e is pretty much the same thing as showing that lim h->0 (e^h - 1)/h = 1. Think about it, one follows pretty easily from the other.
 
  • #12
uart: Thank you. I agree that one follows from the other. Radou's limit, as he's tackling the general case of differentiating a^x, is of (a^h - 1)h, and comes to ln a; but this can also easily be shown to follow from lim h -> 0 (1 + h)^1/h. My point, though, is that lim h -> 0 (1 + h)^1/h arises naturally in the first principles differentiation of log x (base a) and is, arguably, the standard definition of e.
 

What is "D/dx first principles"?

"D/dx first principles" refers to the method of finding the derivative of a function using the limit definition of the derivative. This involves taking the limit as the change in the input variable approaches zero.

Why is it important to understand "D/dx first principles"?

Understanding "D/dx first principles" allows us to find the derivative of any function, even if it is not a simple polynomial. This is a fundamental concept in calculus and is essential for solving complex problems in physics, engineering, and other fields.

How do you use "D/dx first principles" to find the derivative of a function?

To use "D/dx first principles" to find the derivative of a function, we first write out the limit definition of the derivative. Then, we plug in the function and simplify the expression as much as possible. Finally, we take the limit as the change in the input variable approaches zero to find the derivative.

Are there any limitations to using "D/dx first principles" to find derivatives?

One limitation of using "D/dx first principles" is that it can be a time-consuming process, especially for more complex functions. Additionally, it may not be possible to find the derivative of certain functions using this method, as the limit may not exist or may be difficult to evaluate.

How does "D/dx first principles" relate to other methods of finding derivatives?

"D/dx first principles" is the most basic method for finding derivatives and is the foundation for other methods, such as the power rule, product rule, and chain rule. It allows us to understand the concept of a derivative and serves as a starting point for more advanced techniques.

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