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2^x derivitiveby hackensack
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#1
May1904, 04:10 PM

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I need to find the derivitive of y=2^x using the definition of derivitive.



#2
May1904, 04:30 PM

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What have you done so far? What does the definition of derivative say?



#3
May2104, 05:28 AM

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This was also posted in the calculus section and there are about 10 replies there.



#4
Sep2104, 08:27 PM

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2^x derivitive
I doubt this is a suitable problem for a novice. even showing convergence is tough. i will look at the other posted answers. there is a good reason people start from the integral definition of ln(x) to derive this result.



#5
Sep2104, 09:17 PM

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If f(x)= a^{x}, the f(x+h)= a^{a+x}= a^{x}a^{h} so
f(a+ h) f(a)= a^{x}(a^{h} 1). The derivative is lim (f(x+h) f(x))/h= a^{x}lim {(a^{h}1)/h}. Notice that that is a^{x} time a limit that is independent of x. That is, as long as the derivative exists, it is a^{x} times a constant. The problem is showing that the lim{(a^{h}1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2). Showing that that limit exists is sufficiently nontrivial that many people (myself included), as mathwonk said, prefer to define ln(x) as the integral, from 1 to x of (1/t)dt. From that, it is possible to prove all properties of ln(x) including (trivially) that the derivative is 1/x. Defining e^{x} as the inverse function of ln(x) leads to all the properties of e^{x} (including the fact that it is some number to a power!), in particular that its derivative is e^{x} itself and, from that, that the derivative of a^{x} is (ln a) a^{x}. 


#6
Oct1504, 07:23 AM

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dunno if i'm missing the point here but...
write y=2^x as y=exp(x.ln2) => y'=ln2.exp(x.ln2) 


#7
Oct1604, 02:53 AM

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No, you did exactly what HallsofIvy was advocating, he was just pointing out that the question asked for it to be solved using the definition of a derivative, which makes things much harder. Easier to approach things from the other way, starting by defining the integral of 1/x.



#8
Oct2904, 12:11 PM

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"The derivative is lim (f(x+h) f(x))/h= axlim {(ah1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."
tell me if i'm wrong, but it doesn't seems so hard to determine this limit.. (a^h1)/h = (exp (h*ln(a) )1) / h = ( 1 + h*ln(a) + o(h*ln(a))  1 ) / h h>0 = ln(a) + o(ln(a)) so lim (a^h1)/h = ln(a) ....... 


#9
Oct2904, 01:32 PM

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