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 July 10th, 2019, 07:28 AM #1 Senior Member   Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8 People prove the derivative of $\displaystyle e^x$, $\displaystyle \ln(x)$ using either the formula of $\displaystyle e = \lim_{x \to 0} (1+x) ^ \frac{1}{x}$ or if they know one of the derivatives $\displaystyle e^x$, $\displaystyle \ln(x)$ they use it to prove the other. My textbook first gave me without proof that $\displaystyle \frac{d}{dx}e^x = e^x$ Then it found the limit of $\displaystyle \ln(x)$ using the chain rule: $\displaystyle f(x) = \ln(x) \Leftrightarrow e^{f(x)} = e^{\ln(x)} \Leftrightarrow e^{f(x)} = x \Leftrightarrow \frac{d}{dx}e^{f(x)} = \frac{d}{dx}x \Leftrightarrow e^{f(x)} \frac{df(x)}{dx} = 1 \Leftrightarrow \frac{df(x)}{dx} = \frac{1}{e^{f(x)} } \Leftrightarrow \frac{d}{dx}\ln x = \frac{1}{e^{\ln x} } \Leftrightarrow \frac{d}{dx}\ln x = \frac{1}{x}$ And then it uses the limit of ln(x) at x=1 as calculated above with the definition of the derivative in order to show that $\displaystyle e = \lim_{x \to 0} (1+x) ^ \frac{1}{x}$ And then others (on the internet) are using $\displaystyle e = \lim_{x \to 0} (1+x) ^ \frac{1}{x}$ to show that the derivative of $\displaystyle e^x$ is $\displaystyle e^x$. Well the circular problem is solved if you can some how find the derivative of $\displaystyle e^x$ without using any of the above. Then all the other derivatives and e as a limit can be proved by using $\displaystyle e^x$. Can you show me the right order? Well the book might have a good reason that it hasn't shown to me yet; maybe it's really difficult to calculate the limit of $\displaystyle e^x$. Thank you. Last edited by skipjack; July 10th, 2019 at 11:06 AM. July 10th, 2019, 08:32 AM #2 Senior Member   Joined: Oct 2009 Posts: 883 Thanks: 340 If you want to avoid circular reasonings, you'd need to start by adressing what is the DEFINITION of e, e^x and log(x). July 10th, 2019, 08:37 AM   #3
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 Originally Posted by Micrm@ss If you want to avoid circular reasonings, you'd need to start by adressing what is the DEFINITION of e, e^x and log(x).
You mean the infinity series that give e and e^x? July 10th, 2019, 08:56 AM   #4
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 Originally Posted by babaliaris You mean the infinity series that give e and e^x?
I usually see it as

$\ln(x) = \displaystyle \int_1^x~\dfrac{dt}{t}$ July 10th, 2019, 09:06 AM   #5
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 Originally Posted by romsek I usually see it as $\ln(x) = \displaystyle \int_1^x~\dfrac{dt}{t}$
hmm. But still the reason I know how to solve this integral is because I ask my self "What do i need to derive from to get 1/t?" so how does that integral make sense as a definition? July 10th, 2019, 09:30 AM   #6
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 Originally Posted by babaliaris hmm. But still the reason I know how to solve this integral is because I ask my self "What do i need to derive from to get 1/t?" so how does that integral make sense as a definition?
A bit of reading on the web will provide you with all the answers to this question that exist. July 10th, 2019, 11:38 AM   #7
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 Originally Posted by babaliaris . . . how does that integral make sense as a definition?
One needs to start somewhere. For $x > 0$, it defines a function of $x$ that can be called "$\ln(x)$" and has the properties you'd like for a logarithm.

One can now define $e$ as the value of $x$ such that $\ln(x) = 1$. Tags circular, dependency, limit, lnx, proof Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post briankymely Linear Algebra 6 December 14th, 2017 04:12 AM ElectroMan Calculus 1 June 30th, 2015 05:19 AM savvasathanasiadis Linear Algebra 1 February 17th, 2015 09:37 AM vyoung831 Linear Algebra 1 December 12th, 2014 08:20 PM greg1313 Calculus 2 January 19th, 2009 03:56 PM

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