Derivative of a^x.

• Nov 15th 2013, 02:40 PM
sepoto
Derivative of a^x.
Attachment 29739

$M(a)=\frac{\lim}{\Delta x\rightarrow 0}\frac{a^\Delta ^x-1}{\Delta x}$
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$M(a)=\frac{d}{dx}a^x...at...x=0$

"2. Geometrically, M(a) is the slope of the graph y = ax at x = 0."

My question is what about the other points other than 0 where a^x is equal to 1? Right now I think that the slope at 2 is M(a)a^x however the documents seem to me right now to be saying that M(a) only is the slope at the other points such as 2. I'm trying to clear this up.

• Nov 15th 2013, 03:09 PM
Plato
Re: Derivative of a^x.
Quote:

Originally Posted by sepoto
$M(a)=\frac{\lim}{\Delta x\rightarrow 0}\frac{a^\Delta ^x-1}{\Delta x}$
-----------
$M(a)=\frac{d}{dx}a^x...at...x=0$

${\lim _{h \to 0}}\frac{{{a^{x + h}} - {a^x}}}{h} = {a^x}{\lim _{h \to 0}}\frac{{{a^h} - 1}}{h}$

You need to know ${\lim _{h \to 0}}\frac{{{a^h} - 1}}{h} = \log (a)$
• Nov 15th 2013, 08:16 PM
sepoto
Re: Derivative of a^x.
I see that is correct and if I use my calculators ln function that ln(2)*2^3 gives me the derivative of 2^x at the point x=3 for the function 2^x.

What I'm still trying to understand is that if the limit is for the point 0 it looks like it would result in a divide by zero error if the closest point 2^0 is entered:

$\frac{2^0-1}{0}$

which does not seem to be coming out to:

0.6931

which is the actual derivative of 2^x at x=0.

P.S.

O.K. Since it's the limit so I would have to plug in a value as close to zero as I can without actually being on zero.
• Nov 16th 2013, 07:15 AM
HallsofIvy
Re: Derivative of a^x.
You seem to be missing the whole point of a "limit". The only time you can find $\lim_{x\to a} f(x)$ by calculating f(a) is if the function, f, is "continuous at x= a". Because we define "f is continuous at a" by " $\lim_{x\to a} f(x)= f(a)$, that would be circular reasoning except that we have other ways of determining if functions are continuous.

However, even in this case we do NOT take a limit by "plug in a value as close to zero as I can without actually being on zero". There is NO non-zero number closest to 0. We do NOT find limits by "plugging in" values. I don't have time to give you a course in limits here so I suggest you review them in your text book. They are crucial to Calculus and not as trivial as you seem to think.