
New Users Post up here and introduce yourself! 
 LinkBack  Thread Tools  Display Modes 
December 7th, 2017, 03:45 PM  #1 
Newbie Joined: Dec 2017 From: USA Posts: 2 Thanks: 0  Demonstration that the derivative of ln equals 1/x
Hi! Nice to meet you all <333 I would be tremendously grateful if you could possibly give an answer to my questions and help me with this. I have two questions: first of all, what is the limit of an exponential function as x approaches 0? Is it 1? Second one: how exactly could I calculate lim as h approaches 0 of (1+h/x)^(1/h), that somehow must equal e^(1/x)? I am trying to follow the demonstration of why the derivative of ln is the reciprocal function and one step that is unclear is that limit, that is also written as lim of h as it approaches 0 of (1+h)^(1/xh). I tried to make a conjecture using the binomial formula, but I could not find anything (I am honestly not brave to use it for 1/x as an exponent). The only thing I am aware of is that lim of x as it approaches 0 of (1+x)^(1/x)=e. Thank you very much! Hope you can help me. Fondly, ap calc student. 
December 8th, 2017, 02:29 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  Quote:
Quote:
Personally, I prefer to do what many modern Calculus texts do: define "ln(x)" by $\displaystyle \ln(x)= \int_1^x \frac{1}{t} dt$. Then it immediately follows, from the 'fundamental theorem of Calculus', that $\displaystyle \frac{d \ln(x)}{dx}= \frac{1}{x}$. Now define $\displaystyle e^x$ to be the inverse function to $\ln(x)$. That is, if $\displaystyle y= \ln(x)$ then $\displaystyle x= e^y$. So $\displaystyle \frac{dy}{dx}= \frac{1}{x}$ becomes $\displaystyle \frac{dx}{dy}= x= e^y$. That is, $\displaystyle \frac{de^y}{dy}= e^y$ Last edited by skipjack; December 8th, 2017 at 10:37 AM.  
December 8th, 2017, 03:30 AM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 435 Thanks: 247 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
1. Compute $f(\frac{1}{x})$ using the change of variable $u = \frac{1}{s}$ as \[f(\frac{1}{x}) = \int_1^{\frac{1}{x}} \frac{1}{s} \ ds = \int_1^x \frac{1}{u} \ du = f(x) \] 2. Compute $f(xy)$ using the change of variable $u = \frac{s}{y}$ combined with the result from 1 \[f(xy) = \int_1^{xy} \frac{1}{s} \ ds = \int_{\frac{1}{y}}^x \frac{1}{u} \ du = f(x)  f(\frac{1}{y}) = f(x) + f(y) \] A bit more arguing which I'll omit can show that this is the only such function up to scalar multiples which correspond to the base of the logarithm (i.e. $f(x)$ is exactly $\log(x)$ for some base. Finally, one can solve $f(x) = 1$ using quadrature to obtain \[\int_1^x \frac{1}{s} \ ds = 1 \implies x = \lim_{n\to \infty} \left(1+\frac{1}{n} \right)^n = e\] proving that $f$ is the base $e$ logarithm. This might not be the route the OP is taking in AP calculus. I would guess the method of derivation there is to define $\ln$ as the inverse of the exponential which means that $e^{\ln(x)} = x$ for all $x$. Then apply the chain rule \[\frac{d}{dx} \left(e^{\ln(x)} \right) = \frac{d \ln(x)}{dx}e^{\ln(x)} = x\frac{d \ln(x)}{dx} \] Taking a derivative on the RHS and dividing $e^{\ln(x)} = x$ gives \[\frac{d \ln(x)}{dx} = \frac{1}{x} \] Last edited by skipjack; December 8th, 2017 at 10:40 AM.  
December 8th, 2017, 04:40 AM  #4 
Senior Member Joined: Dec 2015 From: Earth Posts: 238 Thanks: 27 
If you can't remember derivative of any function, use inverse function as below: $\displaystyle y'=dy/dx=(dx_{y}/dy)^{1}=1/x'_{y}$ $\displaystyle (\ln x)'=1/(e^{y})'=1/e^{y}=1/x$ Last edited by skipjack; December 8th, 2017 at 10:14 AM. 
December 8th, 2017, 06:38 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra 
Given $$\lim_{x \to 0} (1+x)^{\frac1x} = e$$ we determine $$\lim_{x \to 0} (1+ax)^{\frac1x} = e^a$$ by writing $$\begin{align*}\lim_{x \to 0} (1+ax)^{\frac1x} &= \lim_{x \to 0} (1+ax)^{\frac{a}{ax}} \\ &= \lim_{x \to 0} \left((1+ax)^{\frac{1}{ax}}\right)^a\end{align*}$$ for sufficiently small $x$, we have $1+ax \gt 0$ and so the expression is continuous and we can write $$\begin{align*}&& \lim_{x \to 0} (1+ax)^{\frac1x} &= \left( \lim_{x \to 0} (1+ax)^{\frac{1}{ax}}\right)^a \\ &\text{$x \to 0 \iff ax \to 0$ so} &&= \left( \lim_{ax \to 0} (1+ax)^{\frac{1}{ax}}\right)^a \\ &\text{writing $t=ax$} &&= \left( \lim_{t \to 0} (1+t)^{\frac{1}{t}}\right)^a \\ &&&= (e)^a = e^a\end{align*}$$ In your case, you have a variable $h$ and a constant $x$. Last edited by v8archie; December 8th, 2017 at 06:41 AM. 
December 8th, 2017, 10:46 AM  #6 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,431 Thanks: 105 
$\displaystyle y=\log_{a}x\\ \frac{\Delta y}{\Delta x} = \frac{1}{\Delta x}\log_{a}(1+\frac{\Delta x}{x})=\frac{1}{x}\log_{a}(1+\frac{\Delta x}{x})^{\frac{x}{\Delta x}}\\ \lim_{k\rightarrow 0}(1+k)^\frac{1}{k}= e\\ \frac{dy}{dx} = \frac{1}{x}\log_{a}e\\$ or $\displaystyle e^{\log x}=x\\ e^{\log x}\frac{d\log x}{dx}=1 $ Last edited by skipjack; December 8th, 2017 at 12:17 PM. 
December 21st, 2017, 06:19 AM  #7 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  

Tags 
1 or x, calculus, demonstration, derivative, equals, euler's number, limits, log/ln 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Help with a demonstration  Fernando89  PreCalculus  5  June 27th, 2017 07:11 PM 
demonstration  hadjer2015  Calculus  1  April 14th, 2015 04:15 PM 
A linear function equals its derivative at every point  king.oslo  Linear Algebra  1  June 20th, 2014 05:33 AM 
Showing limit equals derivative  jpanderson8  Calculus  1  September 26th, 2013 09:45 AM 
Help with a demonstration  cartesy  Number Theory  1  December 31st, 1969 04:00 PM 