can anyone please prove that

lim (e^x-1)/x =1
x-o

or
derivative of e^x w.r.t. x is e^x

would really appreciate it....

prettty simple.

take the derivative w.r.t. x then approximate the limit value. ie. substitute 0 for x,result will be 1.

as far as derivative of e(x) w.r.t. x is concerned, express e(x) in terms of trigonometric function (sine and cosine) then take the derivative of those. result should be same


HTH
Loote

loote ko method le kaam garlaa jasto chhaina hai, gaaf dine tesai? simple bhaye pat le kina tasthyo?

Tp,

Put your 10" thick glass on and read the question and then read my answers carefully. i assume you have taken mathematics in the high school even if you are not a math major.



i did those kinda stuffs in my high school back in Nepal (Xavier's). hence the simplicity. I guess Pat must be a high school student or a freshman.


Loote

looote, see below:

d/dx[(e^x-1).(x^{-1})] = [(e^x-1).(-1).(x^{-2})] + [(e^x).(x^{-1})]

= [e^x(x-1) + 1]/x^2

x --> 0 ==> [e^x(x-1) + 1]/x^2 --> 0/0

if you think just taking derivative and then replacing x by 0 works, point out the step where i am wrong.

TP

YOu should take the derivative of the numerator and then divide it by the derivative of the denominator. lim(x tends to zer) (d(e^(x)-1)/dx)/(d(x)/dx)
= lim (x tends to zero) e^(x)=1


seems like you have "polera khaing" calculus....hehehe


no offence though

Loote

ewwwwwwwwwwwww math eats my brain out, ahh!

hey loote, how can you take derivatives of numerator and denominator saparately when there is a single expression, which is a quotient of two expressions? i think you should use quotient rule of derivatives and apply to the whole expression and take limit.

Tp,

If you really have taken Calculus at any point in the past then surely you must have heard two terms:


1) Indeterminate

2) L Hospital's Rule

THe question of Pat is a classic example where if we directly limit the given function it will result in 0/0 which is INDETERMINATE. hence to find the solution we take the derivative of the numerator and that of the denominator and then impose the limit.

Hope i clarified.
Loote

if lim (x tends to zero) f(x)/g(x) results in 0/0 (indeterminate) then

using L HOSPITAL'S RULE

lim (x tends to zero) f'(x)/g'(x) will give the result.


Loote

loote, you should have reminded me with the L'Hospital Rule at the begining. 10 barsha bhai sakechha calculus nahereko pani.

u guys kidding me..... 11 wata reply bhayo kati jana le kati wata method bata prove gardi sake hola...wah mera nepali daju bhai haru bhanhya ta autai kurama kichkich garera basera.........
its just for a simple calculus class...and i don't think we have yet covered
1) Indeterminate
2) L Hospital's Rule
i just want a simple proof that says derivative of e^x w.r.t x is e^x ...don't bother about the other one......

loote's got the answer
ok....so can someone prove l'hôpital's rule?

An interesting fact about l'hôpital:

His name was "The Marquis de l'hôpital". The "^" wasn't in use in France untill after the French revolution. Thus before that, Mr "The Marquis" would have written his name as:

l'Hospital.
Since he wrote the first book on calculus ever, even though Newton and Leibniz discovered it separately, imagine the amazement on American's and Englishmen's faces when they mustve first seen his name on the calculus text.

well, a majority of you may not find it funny.

oh and pat as far as derivative of e^x is e^x is concerned, it is probably in your book. look around for a proof in the "logarithms" section of your calculus book. Seems like this is the first time you're taking calculus, ive taken 4 of them already so believe me i know quite a bit. So go earch in your book. You'll find a proof that starts with a logarithmic function and a few graphs to help you understand. A rigorous proof without graphs is nearly impossible.

Hope that helps

P.S. NO UNIVERSITY will even ask you that as a proof in any exam or quiz. If your prof does you can literally hang a whole row of malta-bomb on his manhood and light them.
Use gamala, alu bomb whatever. But if you're wondering HOW TO DO IT as a result of your own inquisitive nature then go look in your book.

thats a very nice suggestion time traveller... and guess what...i have already checked the book..it doesn't have one..
my prof has given me the proof and also said that it won't be tested...he said that he is trying to figure out a basic proof so i thought i'll try and help him with the help of you guys...that's it....
So anyone?

U guys kidding me..... 11 wata reply bhayo kati jana le kati wata method bata prove gardi sake hola...wah mera nepali daju bhai haru bhanhya ta autai kurama kichkich garera basera.........
========================

Pat, pat, pat!!! NO one is kidding here. U ASKED A QUESTION AND WE ARE HERE TO HELP YOU OKAY? AND DON"T THINK WE ARE THE SITE MODERATORS SO WE MUST ANSWER YOU ANYWAYS.

If someone is wondering why there is a word called "smarta$$" in dictionary, then it is to use upon guys like U!!

1) Look at my first post, i have answered all your queries.

2) If you don't know, just say u don't know. no need to pop up later and divert the question. there is nothing wrong in not knowing. no one knows everything.

3) I hope you got the L Hospital's thing.

4) Regarding the proof of e^x thing is concerned there are number of ways.

a) proof using classical rule will be tedious but is found in the appendices of most of the books in calculus as TT pointed out.

b) express e^x in terms of sine and cosine (AS I SAID IN MY FIRST POST), and then impose the limit on those.

c) start with d(ln(x))/dx=1/x, then substitute x by e^x then continue solving, u will get the result.

d) there could be other methods too for solving this.

Loote

Loote ra Thaha Payen kasto NERDY. online basera hisaab solve garidiney...hmmm :P:P

Here is a handout he gave it to me..................
he gave the complicated proof of e^x
I quote
" If the above proof seems intricate, consider the simpler but inexact "proof" below which is found in some calculus textsh...
Per the definition of derivative:
derivative of e^x =lim (e^(x+h)-e^x)/h = e^x lim (e^h-1)/h
h-o
now consider the heuristic argument that says for small h , e^h ~~ 1+h=h ( ~~ means "approx")
substituting we get
derivative of e^x = e^x lim ((1+h)-1)/h =e^x lim h/h = e^x

although tha above reasoning is appealing , consider the hidden flaw.......
we know derivative of a^x where a is constant =a^x ln a
but using the same method above we get,
derivative of a^x = a^x which is not true.."


4) Regarding the proof of e^x thing is concerned there are number of ways.

well loote if you can prove it then stop bi+ching around and just do the proof...I dont want any useless theory , speculations, smart a$$ comments ... just the proof

Pat, you should have thanked Loote at least for the first problem. He did the complete proof. And, those 'Indeterminate' and 'L Hospital's Rule ' are basic things when you start solving Limit problem. And, for the second one, I hope taking Sine and Cosine should help, I really could not remember how it goes.

L'hopital's rule is handy, mind you Pat!