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May 13th, 2010, 05:00 PM  #1 
Senior Member Joined: Mar 2010 From: Melbourne Posts: 178 Thanks: 0  1st and 2nd Derivatives ..
HI, I have: e^(x) / x I need to find the 1st and 2nd derivatives.  My first question is, when finding the first derivative, I converted it from the quotient representation to the product representation for ease. e^(x)/x == e^(x)(x^(1/2)) Finding the derivative of this (using the product rule), I have: (e^(x)(x^1)) + (e^(x)1x^(1/2)) = (e^(x)(x^1))  1(e^(x)x^(1/2)) = (e^(x)(x^1))  (e^(x)x^(1/2)) = (e^(x)/(x))  (e^(x)/x^(1/2)) Now, Wolfram Alpha is reporting: = (e^(x)/(x))  (e^(x)/x^(2)) ... how does (e^(x)x^(1/2)) become: (e^(x)/x^(2)) ?? wulfgarpro. 
May 13th, 2010, 05:40 PM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: 1st and 2nd Derivatives ..
Something's fishy... (e^x)/x = (x^1)(e^x) So you can differentiate this using the product rule... (x^1)` *(e^x) + (x^1)(e^x)` (I'm using ` to represent the derivative...) (1x^2)(e^x) + (x^1)(e^x) (1x^2)(e^x) + x(x^2)(e^x) (Multiply and "divide" by x to get a "common denominator", of sorts) factor out... (e^x)(x^2)[1 + x] Got it from here? 
May 13th, 2010, 06:59 PM  #3 
Senior Member Joined: Mar 2010 From: Melbourne Posts: 178 Thanks: 0  Re: 1st and 2nd Derivatives ..
Wow .. I lost the plot. Would be great if someone could confirm my ANS: Find the 2nd derivative of e(^x)/x.  e(^x)/x is equivalent to: e^(x)x^(1) Using the product rule: (x^(1)e^(x)) + (1x^(2)e^(x)) = e^(x)/x  e^(x)/x^(2) :: 1ST Derivative  Now, finding the 2nd derivative: e^(x)/x  e^(x)/x^(2) is equivalent to: (e^(x)x^(1))  (e^(x)x^(2)) Using the product rule: (e^(x)x^(1)) = (e^(x)x^(1)) + (e^(x)1x^(2)) = (e^(x)/x)  (e^(x)/x^(2)) (e^(x)x^(2)) = (e^(x)x^(2)) + (e^(x)2x^(3)) = (e^(x)/x^(2))  (2e^(x)/x^(3)) Adding back these derivatives back into the 1st derivative we have: = (e^(x)/x)  (e^(x)/x^(2))  (e^(x)/x^(2))  (2e^(x)/x^(3)) = (e^(x)/x)  (2e^(x)/x^(2))  (2e^(x)/x^(3)) :: 2ND Derivative  wulfgarpro. 
May 13th, 2010, 07:06 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  Re: 1st and 2nd Derivatives ..
Everything looks good to me except (e^(x)/x)  (2e^(x)/x^(2)) + (2e^(x)/x^(3)).

May 13th, 2010, 08:39 PM  #5 
Senior Member Joined: Mar 2010 From: Melbourne Posts: 178 Thanks: 0  Re: 1st and 2nd Derivatives ..
Hm, From: (e^x)(x^1)+(e^x)(1x^2)(e^x)(1x^2)+(e^x)(2x^3) I know the correct answer is: {e^x} / {x}  {e^x} / {x^2}  {e^x} / {x^2} + {2e^x} / {x^3} But from what I can see, does not (e^x)(1x^2) become +(e^x)(1x^2) ? wulfgarpro. 
May 13th, 2010, 08:44 PM  #6 
Senior Member Joined: Nov 2008 Posts: 265 Thanks: 0  Re: 1st and 2nd Derivatives ..
Using the quotient rule, I get [e^x(x^2  2x + 2)]/x^3 for the second derivative. In situations like this, I've always found it easier to use the quotient rule for some reason. 
May 13th, 2010, 08:49 PM  #7 
Senior Member Joined: Mar 2010 From: Melbourne Posts: 178 Thanks: 0  Re: 1st and 2nd Derivatives ..
I'm so close though. One sign out. 0.o. wulfgarpro. 
May 13th, 2010, 09:15 PM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  Re: 1st and 2nd Derivatives ..
If it's any help: 
May 13th, 2010, 09:50 PM  #9 
Senior Member Joined: Mar 2010 From: Melbourne Posts: 178 Thanks: 0  Re: 1st and 2nd Derivatives ..
Hm, Still cant see how it works. From my answer: (e^x)(x^1)+(e^x)(1x^2)(e^x)(1x^2)+(e^x)(2x^3) Broken into two: (e^x)(x^1)+(e^x)(1x^2) (i) '' (e^x)(1x^2)+(e^x)(2x^3) (ii) (i) (e^x)(x^1)  (e^x)(x^2) (ii) (e^x)(x^2) + (2e^x)(x^3)  Rejoining them: (e^x)(x^1)(e^x)(x^2)(e^x)(x^2)+(2e^x)(x^3) (e^x)(x^1)(e^x)(x^2)+(e^x)(x^2)+(2e^x)(x^3)  Which does not work out ? Any idea where I'm going wrong ? wulfgarpro. 
May 13th, 2010, 10:22 PM  #10 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond  Re: 1st and 2nd Derivatives ..
Start with: (e^x)(x^1) Differentiate: (e^x)(x^1)(e^x)(x^2)  differentiate again: (e^x)(x^1)(e^x)(x^2)  ((e^x)(x^2)  (e^x)(2x^3)) Watch the signs! 

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