anonymous one year ago Can anybody explain the steps to the algebra that arrives at the derivative function to f[x] = x/e^(x^2) f'[x] = ???

1. Australopithecus

are you using the definition of the derivative or are you allowed to use the rule set?

2. logan13

35

3. anonymous

I think at this point anything goes, I'm mainly just trying to understand how the f[x+h] - f[x] /h equation cancels out.. or if there is another alternative approach that gets me there. Its not actually for any test.. just my ability to use and apply it.

4. logan13

ya me 2

5. Australopithecus

Here is a list of basic calc rules (calculus 1) If you want to solve the derivative quickly I would suggest learning the rules they are included on this pdf. I can help you use them and break it down for you if you want. Although if you want we can work through the definition of the derivative method, been a long time since I have dont it but shouldn't be too hard.

6. anonymous

okay I think that means the definition of the derivative for me

7. anonymous

ah thats awesome.. man that's a lot of rules.. geez, how do you guys memorize this?

8. Australopithecus

Its actually not that hard just like learning algebra rules. YOu apply them a million times and you can just look at a function and see its derivative

9. anonymous

yeah, I can see how this would save a lot of fiddly algebra

10. Australopithecus

So first you need to set up the expression do you know how to? Hint: for f(x+h) you just replace all x terms in the function with (x+h)

11. anonymous

yes 1 sec .. I can post how far I got

12. Australopithecus

Doing the definition of this will probably be really dirty actually lol

13. Australopithecus

14. anonymous

$\left( \frac{ x+h }{e^{(x+h)^{2}}} - \frac{x}{e^{x^{2}}} \right)* \frac{1}{h}$ $\left( (x+h)e^{-(x+h)^2}-x e^{-x^{2}} \right) * \frac{1}{h}$ $\frac{ (x+h)e^{-(x+h)^2}-x e^{-x^{2}} } {h}$ and a few other variations of that

15. anonymous

thanks I really really appreciate it.. this is going on 14 hours of head scratching for me now

16. anonymous

Im not sure if I want to keep a denominator or change it somehow, or if there is a good way to get rid of it.

17. hartnn

you've solved limits questions before? have you come across $$\Large \lim \limits_{x\to 0 } \dfrac{e^x -1}{x}$$ ?

18. anonymous

theres a good question ... hmmm ... its been a while.. about a year, but I think I did that in my high school pre calc..

19. hartnn

good, remember L'Hopital's rule? that can be easily solved by it...

20. anonymous

I couldnt tell you off hand how to solve that..

21. anonymous

I can go look it up and refresh though

22. hartnn

by L'Hopital's rule $$\Large \lim \limits_{x\to 0 } \dfrac{e^x -1}{x} =\Large \lim \limits_{x\to 0 } \dfrac{\dfrac{d}{dx}[e^x -1]}{\dfrac{d}{dx}[x]} = \\ \Large \lim \limits_{x\to 0 } e^x = e^0 =1$$ If you haven't learnt how to take the derivatove, you can always take $$\Large \lim \limits_{x\to 0 } \dfrac{e^x -1}{x} = 1$$ as a standard result

23. hartnn

now with the algebra part, $$\Large (x+1) e^{-(x+1)^2 } - xe^{-x^2}$$ expand (x+1)^2 if we look carefully, we can just factor out $$e^{-x^2}$$ from that can you try that ?

24. anonymous

Yeah, I dont think I've covered this before.. but.. it looks like it makes some sense.. there are two opposing functions of some kind that are equal at extreme infitesimal values of x and therefore give the result one. And from this I guess we can simplify the equation. I'll go read up and fill in this blank in my understanding.

25. hartnn

x+h, not x+1 sorry :P

26. hartnn

i'll type the simplification by the time you try $$\Large (x+h) e^{-x^2 +2xh-h^2 } - xe^{-x^2}$$ $$\Large (x+h) e^{-x^2} e^{2xh-h^2 } - xe^{-x^2}$$ $$\Large e^{-x^2} [(x+h) e^{2xh-h^2 } - x]$$

27. anonymous

ahhh! you expanded the (x+h)^2 then used the e^(a + b) = e^a e^b rule to get to stage 2.. then factored out e^-x^2 .. jeez I was having a hell of time working out how to do that.. I got basic algebra holes

28. hartnn

thats right! :) now forget about e^(-x^2) that will be a constant and will be pulled out of the limit, as we are taking the limit with "h" as variable. $$\Large [(x+h) e^{2xh-h^2 } - x]= x(e^{2xh-h^2 }) + h (e^{2xh-h^2 } )-x$$ now we have 3 terms and don't forget the h in the denominator

29. hartnn

the middle term is interesting $$\huge \dfrac{h(e^{2xh-h^2})}{h} = e^{2xh-h^2}$$ since h->0 , you can directly plug in h =0 in that ^^

30. hartnn

from the other 2 terms, factor out the x $$\Large \dfrac{x(e^{2xh-h^2 }) -x}{h}= x[\dfrac{(e^{2xh-h^2 }) -1}{h} ]$$ now doesn't that fraction look very similar to e^x-1 ;) I'll let you try further... will come later to check how much you could solve :)

31. hartnn

ask if you have any doubts in any explanation above

32. anonymous

I will. thanks, just digesting it..

33. anonymous

Im having trouble finishing this with a working function.. One of the problems I have is when I go back to this point.. I dont get the same result when I expand.. instead I get .. $\Large \left((x+h) e^{-(x+h)^2 } - xe^{-x^2}\right) * \frac{1}{h}$ $\Large \left((x+h) e^{-((x+h)(x+h)) } - xe^{-x^2}\right) * \frac{1}{h}$ $\Large \left( (x+h) e^{-x^2-2hx-h^2 } - xe^{-x^2}\right) * \frac{1}{h}$ $\Large \left( x e^{-x^2-2hx-h^2 } + h e^{-x^2-2hx-h^2 } - xe^{-x^2}\right) * \frac{1}{h}$ $\Large e^{-x^2} \left( x e^{-2hx-h^2 } + h e^{-2hx-h^2 } - x\right) * \frac{1}{h}$ And before I go any further may I confirm if I haven't got it wrong from the start?

34. Australopithecus

Ok so I solved this problem I didnt use L'Hopital's rule. Not really following how the other guy is using it. This is a pretty weird problem, was kind of fun solving because the weird limit at the end. First off to solve this problem you need to know limit rules Here is a list of them http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/limit_laws.html First off, your algebra looks fine. Step 1 cancel out h terms in your equation Step 2 you should notice that one of your terms does not have h in the numerator or denominator thus you can treat it as its own function and apply this rule called the addition law: $\lim_{h \rightarrow a} (g(x) + f(x)) = \lim_{h \rightarrow a} g(x) + \lim_{h \rightarrow a} f(x)$ Also notice that you can treat x terms such as e^-x^2 and x as constants so thus you can simplify your problem even more using the rule called the constant law: $\lim_{h \rightarrow a} c*f(x) = c*\lim_{h \rightarrow a} f(x)$ (in this case c would be x and e^-x^2 terms) Step 3 The term with no h in it you can solve its limit leaving you with only two terms, combine those terms factor out all the x terms you can. Then you will be left with a pretty weird limit, but it is solvable. Step 4 once you have the weird limit you are going to have to do some funky math tricks (completing the square and realizing and factoring out a term) to get it in the form of the definition of a derivative from there you can realize that the solution to the limit is the derivative of a specific exponential function. I just used derivative rules to solve the limit but you can use the definition. the link below goes over how to solve exponential functions using the definition of a derivative if you need help let me know: http://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx If you need help with this final step let me know and if you are confused with any of the details I would be happy to walk you through it.

35. Australopithecus

Give me a bit I need to look up a proof for the exponential rule using definition of a derivative

36. Australopithecus

or you could just be content with d/dx e^(f(x)) = f'(x)e^(f(x))

37. Australopithecus

Do you follow my steps though? they are kind of arbitrary but yeah

38. anonymous

thanks, I can see how I can get rid of one h easily.. in the middle term.. the other two, I'm not sure how easily they will go. $\Large e^{-x^2} \left(\frac{ x e^{-2hx-h^2 }}{h} + \frac{ h e^{-2hx-h^2} }{h} - \frac{x}{h} \right)$ $\Large e^{-x^2} \left(\frac{ x e^{-2hx-h^2 }}{h} + \frac{ e^{-2hx-h^2} }{1} - \frac{x}{h} \right)$ I feel like I need to do something about getting rid of those exponents already, do you have a trick for reducing xe^... any more ?

39. Australopithecus

You need to use the limit notation while solving your problem

40. Australopithecus

so right now, $\lim_{h \rightarrow 0} e^{-x^2}(\frac{xe^{-2hx-h^2}}{h} + e^{-2hx-h^2} + \frac{x}{h})$

41. Australopithecus

now apply the rules I showed you above

42. anonymous

cool this is looking good, okay I might need a bit to read up on pauls notes there.. and get the gist of how the rules apply on this function as it is.. that might take me a bit.. then I have an appointment at 6 tonight., so it might be a few hours before I have this down.

43. Australopithecus

Lets just try to get it in the right form first?

44. anonymous

feels like it's 1 minute from an ah ha! moment though

45. Australopithecus

then we can try to tackle the limit

46. Australopithecus

so yeah using proof to solve the limit is apparently very long and tricky according to some people who know what they are talking about. the link I posted wont be helpful in this case I dont think

47. anonymous

if I understand it right, this function will become several limit functions?

48. Australopithecus

No you will split this into two functions

49. anonymous

and conceptually, x and e will become roughly the same very small value

50. Australopithecus

you have 3 terms, you split them into two functions, then you solve one of the limits, then you combine the function with two terms and simplify

51. Australopithecus

52. anonymous

...looking at that

53. Australopithecus

If you want an example I can provide one?

54. Australopithecus

For example $\lim_{h \rightarrow 0} (x + \frac{x}{h})$ Applying addition rule, f(x) = x and g(x) = x/h so, $\lim_{h \rightarrow 0} (x + \frac{x}{h}) = \lim_{h \rightarrow 0} x + \lim_{h \rightarrow 0} \frac{x}{h}$

55. Australopithecus

we can also apply the constant rule here by assuming x is a constant because we are taking the limit in terms of h so, $\lim_{h \rightarrow 0} x + \lim_{h \rightarrow 0} \frac{x}{h} = x* \lim_{h \rightarrow 0} 1 + x*\lim_{h \rightarrow 0} \frac{1}{h} = x(\lim_{h \rightarrow 0} 1 + \lim_{h \rightarrow 0} \frac{1}{h})$ this isnt the best example because there is no limit for 1/h thus the addition rule is not valid but yeah just an example of how to do the manipulatons

56. Australopithecus

or rather the limit of x/h does not exist

57. anonymous

ah ok is this where I should be next? $\lim_{h \rightarrow 0} e^{-x^2} (\lim_{h \rightarrow 0} + \frac{xe^{-2hx-h^2}}{h} + \lim_{h \rightarrow 0} \frac{x}{h}+ \lim_{h \rightarrow 0} e^{-2hx-h^2} )$

58. Australopithecus

No sorry that is wrong, lets take a step back, you have three terms, and you want to treat all individual x terms as constants. so first off you want to designate an f(x) term and a g(x) term

59. Australopithecus

look at the first part of my example

60. Australopithecus

you want to group the terms with h in the denominator

61. Australopithecus

using the rule, $\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$

62. Australopithecus

then you will have two terms

63. Australopithecus

Then you can declare them as separate functions and apply the addition rule

64. anonymous

So first if I understand this right, then that term on the front can be ignored, because it does not have h in it, I think this is what you mean by constant? $\lim_{h \rightarrow 0} \frac{xe^{-2hx-h^2 }+x}{h} + \lim_{h \rightarrow 0} e^{-2hx-h^2}$

65. anonymous

or at least I am guessing for the time being, can be put on the side. and used later.

66. Australopithecus

You should have e^(-x^2) multiplied by everything but yeah that is right

67. anonymous

ok, yes, that what I was thinking..

68. Australopithecus

so, $e^{-x^2}(\lim_{h \rightarrow 0} \frac{xe^{-2hx - h^2} +x}{h} + \lim_{h \rightarrow 0} e^{-2hx - h^2})$

69. Australopithecus

You can solve one of your terms leaving you with only one limit to solve

70. Australopithecus

factor out x of the last remaining limit

71. anonymous

$e^{-x^2}(\lim_{h \rightarrow 0} \frac{x(e^{-2hx - h^2} +1)}{h} + \lim_{h \rightarrow 0} e^{-2hx - h^2})$

72. Australopithecus

yeah now you can solve the limit of e^(-2hx - h^2)

73. anonymous

should I be thinking something along the lines of dropping that second part of that exponent.. (my signs might be off) but something like $e^{-x^2}(\lim_{h \rightarrow 0} \frac{x(e^{-2hx - h^2} +1)}{h} + \lim_{h \rightarrow 0} e^{-2hx}/e^{- h^2})$

74. Australopithecus

for the second limit all you have to do is sub 0 into h

75. Australopithecus

the last limit you need to manipulate it to look like the definition of a derivative

76. anonymous

-2x h - h*h = -2x *0 - 0*0 = 0 ? or do think of this as 2x and 0's cancel out in some way ?

77. Australopithecus

no that is right you will have e^0 for the second limit

78. Australopithecus

the reason we cant solve the first limit is because h is in the denominator and dividing by 0 is undefined

79. anonymous

oh cool, seemed too easy

80. Australopithecus

so you have the first term of the derivative but to find the second you have to manipulate the last remaining limit to look like the definition of a derivative, hint complete the square in the exponential and factor

81. anonymous

$e^{-x^2}(\lim_{h \rightarrow 0} \frac{x(e^{-2hx - h^2} +1)}{h} + \lim_{h \rightarrow 0} e^{-2x* 0- 0 * 0})$ $e^{-x^2}(\lim_{h \rightarrow 0} \frac{x(e^{-2hx - h^2} +1)}{h} + \lim_{h \rightarrow 0} e^{0})$

82. Australopithecus

yeah so, you have the lim h->0 e^0 = 1

83. Australopithecus

do you know how to complete the square?

84. anonymous

ah right I just spotted that ... $e^{-x^2}(\lim_{h \rightarrow 0} \frac{x(e^{-2hx - h^2} +1)}{h} + \lim_{h \rightarrow 0} 1)$

85. Australopithecus

you should remove the limit notation from the second one

86. Australopithecus

since it is redundant

87. anonymous

yeah, bit rusty on it.. there's a couple of ways I recall. not sure if these terms divide nicely..

88. Australopithecus

the last limit is a bit tricky

89. Australopithecus

do you have any idea how to make it look like the definition of a deriative

90. anonymous

I have an idea of how it should look when done..

91. Australopithecus

So the last limit you want it in the form of $\frac{f(x+h) - f(x)}{h}$

92. Australopithecus

well when you are done you wll have a function in terms of x that is the derivative you are looking for

93. anonymous

$f'[x] = ​(1 - 2*x^2)* e^{-x^2]}$

94. Australopithecus

huh?

95. anonymous

when I cheat and ask mathematica to give me the derivative of f[x] that's what it gives me.

96. Australopithecus

ok so you have this right now $e^{-x^2}(\lim_{h \rightarrow 0} \frac{e^{-2hx-h^2}-1}{h} + 1)$

97. Australopithecus

yeah that is the derivative notice when you expand you get e^(-x^2) which is showing up in the problem atm

98. anonymous

yes

99. Australopithecus

oops

100. Australopithecus

$\lim_{h \rightarrow 0} \frac{x(e^{-2hx-h^2}-1)}{h}$ notice that x is a constant

101. Australopithecus

this is the limit you have to solve

102. Australopithecus

I forgot to write HINT: notice x is a constant

103. Australopithecus

you need to rewrite the limit so it looks like the definition of a derivative

104. Australopithecus

you can achieve this by using completing the square and factoring

105. Australopithecus

out a term

106. Australopithecus

its actually kind of tricky

107. Australopithecus

then once you have it in that form you can tell what functions derivative is the solution to the limit

108. anonymous

yeah I have a dumb question.. usually I complete the square on a nicely formed quadratic ax^2 + bx + c ... I'm having a bit of problem seeing where that is

109. Australopithecus

You need to complete the square in the exponential

110. Australopithecus

so you need to complete the square with the following: -2hx-h^2

111. Australopithecus

then after you complete the square with the exponential you will need to remember this rule: $e^{a+b} = e^a(e^b)$ and also that if: $e^{a}e^{-a} = e^{0} = 1$

112. Australopithecus

so an example if I had 4ab + 4b^2 I could complete the square by writing (a + 2b)^2 - a^2

113. Australopithecus

This is actually fairly tricky algebra

114. Australopithecus

but not really once you see it you are like duh but yeah took me awhile to figure it out

115. anonymous

Just had to read my high school notes on completing the square again.. (x + b/2)^2 = c ax^2 + bx + c = -h^2 -2h x + 0 = 0 -1 -1 (-h -1x)^2 = 0 I'm having a bit of a brain skid, what to do with that c

116. anonymous

and the x

117. Australopithecus

just make a ( )^2 + what ever needs to get canceled out from the expanded ( )^2

118. Australopithecus

look at my example when you expand (a+2b)^2 you get an extra term so for (a+2b)^2 to equal 4ab + 4b^2 you need to subtract a term from (a+2b)^2

119. Australopithecus

I have totally forgot that formula lol

120. Australopithecus

instead I just think of a square that has the same terms in it then I subtract or add something from it to make it equal to what ever I started with

121. Australopithecus

Here are some videos on completing the square https://www.youtube.com/watch?v=xGOQYTo9AKY https://www.youtube.com/watch?v=Q0IPG_BEnTo maybe these can help I need to go out and eat I will be back later

122. anonymous

man, thanks a lot tagging along on this journey with me and your help and patience yeah, I need a little time to remember this stuff again.. I used to be really good at completing the square and had it totally down.. could see it conceptually.. now Im just drawing a blank.

123. Australopithecus

Well I recommend looking at the example provided it is very similar to yours Here it is again 4ab + 4b^2 = (a + 2b)^2 - a^2

124. Australopithecus

Did you figure it out?

125. anonymous

Where I get stuck on this completing the square problem is that I have this extra x in the equation and the damn negative sign on both terms.. 4ab + 4b^2 "I could complete the square by writing" (a + 2b)^2 - a^2 (a + 2b)(a+2b) a^2+2ab+2ab+4b^2 (a^2+4ab+4b^2) -a^2 ( 4ab+4b^2) okay so GIVEN -2hx -h^2 (2hx+h^2) *-1 can this be factored like this ???? Then (h +1x)(h +1x) (h +x)(h +x) (h^2 +hx)+(hx +x^2) (h^2 +2hx + x^2) -x^2 gives original expression (h^2 +2hx ) -((h+x)^2 - x^2) -(h+x)^2 + x^2

126. Australopithecus

yup

127. Australopithecus

good job

128. anonymous

lol omg.. that was an effort

129. Australopithecus

ok write it out

130. Australopithecus

you are close to it looking like the definition of a specific derivative

131. Australopithecus

you only really have a single manipulation left look at the rules I gave you above for a hint

132. Australopithecus

you need to factor it out of the limit

133. Australopithecus

hint it is the term you added just by completing the square

134. Australopithecus

remember you want it to look like the definition of a derivative when you are done, that way you can find the specific function you need to find the derivative of to solve the limit

135. Australopithecus

look at the two rules I posted regard exponentials they are hints to what you have to do

136. anonymous

$\lim_{h \rightarrow 0} \frac{x(e^{-2hx-h^2}-1)}{h}$ $\lim_{h \rightarrow 0} \frac{x(e^{-(h+x)^2} e^{x^2}-1)}{h}$ Im thinking something like this

137. Australopithecus

that's good You should factor x out of the limit as it is a constant (look at constant rule) You want to factor out e^(x^2) again look at my exponential hints

138. Australopithecus

you want to pull e^(x^2) out of the limit

139. Australopithecus

or exponential rules I gave you I mean

140. anonymous

sorry for delay.. juggling the right exponent law (e^a)^b = e^(a b) . e^(a + b) = e^a e^b

141. anonymous

trying to see how they fit

142. Australopithecus

143. Australopithecus

remember you are trying to make it look like the definition of a derivative

144. anonymous

should I try expanding that (h+x)^2 again ? h^2 +2hx +x^2

145. Australopithecus

no

146. anonymous

I can see how that adds another e^x^2

147. Australopithecus

we are trying to factor out e^(x^2) from the limit, so that we have an equation as so: $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ we are trying to find f(x)

148. Australopithecus

Do you follow?

149. anonymous

not connecting the dots.. you might need to leave me with it for bit.. I'll get it.. I wont give in to the next ah ha moment.. but it might be 1 minute, might be an hour

150. Australopithecus

You will get it, just look at this rule: $e^{a}e^{-a} = e^{a-a} = e^{0} = 1$

151. anonymous

I got lost down this tangent.. looking for a way to factor e^(x^2) out of the first term.. but that dont look right $\lim_{h \rightarrow 0} \frac{x(e^{-(h+x)(h+x)} e^{x^2}-1)}{h}$ $\lim_{h \rightarrow 0} \frac{x(e^{-(h^2+2xh)-x^2} e^{x^2}-1)}{h}$

152. Australopithecus

To factor out e^(x^2), you need to change 1 into something

153. Australopithecus

I gave you a rule that might help

154. anonymous

are we zeroing out h ? here, I can see that creating e^-(0+x)(0+x) = e^-(x^2)

155. Australopithecus

just factor e^(x^2) out of the limit

156. Australopithecus

and you will see what I mean

157. Australopithecus

look at the rule I posted you need to change 1 to something so you can factor out e^(x^2)

158. anonymous

this rule e^{a}e^{-a} = e^{a-a} = e^{0} = 1

159. Australopithecus

right

160. anonymous

and I change the 1 in that rule, or the -1 at the end of the equation ?

161. Australopithecus

we are focusing on the limit

162. Australopithecus

the -1 is not in the limit anymore

163. anonymous

I am staring at this equation .. maybe Im looking in the wrong place $\lim_{h \rightarrow 0} \frac{x(e^{-(h+x)^2} e^{x^2}-1)}{h}$

164. Australopithecus

oh sorry yeah the -1

165. anonymous

so another type of 1... e^(x^2) / e^(x^2) ?

166. Australopithecus

yeah

167. Australopithecus

now you can factor e^(x^2) and x out of the limit and then look at what you have

168. Australopithecus

you still here?

169. anonymous

yes .. staring at this ... $\lim_{h \rightarrow 0} \frac{x(e^{-(h+x)^2} e^{x^2}- e^{x^2} / e^{x^2})}{h}$

170. Australopithecus

well you need to use the constant rule

171. Australopithecus

remove x and e^(x^2)

172. anonymous

wondering how Im going to get that e^x^2 out of there..

173. Australopithecus

you can factor e^(x^2) and x out of the limit look at the limit rules I posted

174. anonymous

lookign them up ...

175. Australopithecus

I will just write it

176. anonymous
177. Australopithecus

$\lim_{x \rightarrow a} cf(x) = c*\lim_{x \rightarrow a} f(x)$

178. Australopithecus

that is the constant rule

179. Australopithecus

in this case x and e^x^2 are constants

180. anonymous

trying to understand how that rule is applied to the equation ...

181. anonymous

does that mean x and e^x^2 are replaced with some value, or cancelled out? or that they now have somer relationship to h in some way?

182. Australopithecus

they dont cancel out they just leave the limit

183. anonymous

it looks like they multiply the value of the original function ...

184. anonymous

oh, so they are allowed to be removed from future calcs.. because they are not actually affected by any change in h

185. Australopithecus

They get pulled outside the limit

186. Australopithecus

yes

187. Australopithecus

so your limit will be simplified

188. anonymous

goodness me.. Im a slow as brick.

189. Australopithecus

its ok you aren't dumb just learning

190. Australopithecus

so write out what you have now

191. anonymous

$\lim_{h \rightarrow 0} \frac{e^{-(h+x)^2} }{h}$

192. Australopithecus

nope

193. anonymous

ok cool

194. Australopithecus

close though

195. Australopithecus

what did you do with e^(x^2), e^(-x^2) and x

196. anonymous

I threw them away.. so they need to stay in the equation somehow, but outside the limit.

197. Australopithecus

yes except one of the terms you need to leave in the limit

198. Australopithecus

e^(-x^2) needs to stay in the equation

199. Australopithecus

but x and e^(x^2) can be factored out

200. anonymous

maybe Im not seeing $\lim_{h \rightarrow 0}$ In the right way.. should I be seeing this as some kind of term ?

201. Australopithecus

?

202. Australopithecus

go back to what you had originally

203. Australopithecus

204. anonymous

$\lim_{h \rightarrow 0} \frac{x(e^{-(h+x)^2} e^{x^2}- e^{x^2} / e^{x^2})}{h}$

205. Australopithecus

yes

206. anonymous

$\lim_{h \rightarrow 0} \frac{x(e^{x^2}( e^{-(h+x)^2} - e^{x^2} ))}{h}$

207. Australopithecus

closer now just pull x and e^(x^2) out front

208. Australopithecus

of the limit

209. Australopithecus

now look at your limit look familure?

210. Australopithecus

hint look at the definition of a derivative

211. anonymous

$x(e^{x^2}( \lim_{h \rightarrow 0} \frac{ e^{-(h+x)^2} - e^{x^2} }{h} ))$

212. anonymous

lol.. thats got to be wrong

213. anonymous

the limit definitely looks the shape we were after

214. Australopithecus

yes but it should be written as: $xe^{x^2}(\lim_{h \rightarrow 0} \frac{e^{-(h+x)^2} - e^{x^2}}{h})$ but you are correction

215. Australopithecus

what does that look like?

216. Australopithecus

hint it is just the definition of a derivative

217. Australopithecus

but what is f(x)?

218. Australopithecus

if you know f(x) you know that f'(x) is the solution to that limit

219. anonymous

like some kind of inversion

220. Australopithecus

what function would you have to plug into the definition of a derivative to get that equation

221. Australopithecus

or that limit rather

222. Australopithecus

Make a guess

223. anonymous

the e^x^2 ?

224. Australopithecus

225. Australopithecus

does e^(x)e^(x) = 1?

226. Australopithecus

by the rule I posted

227. anonymous

needs a negative

228. Australopithecus

yeah

229. Australopithecus

so what is the function then?

230. anonymous

$xe^{-x^2}(\lim_{h \rightarrow 0} \frac{e^{-(h+x)^2} - e^{x^2}}{h})$ i think this is what you are asking for ? $xe^{-x^2}$

231. Australopithecus

that doesnt make sense

232. Australopithecus

you want to factor out e^(x^2)

233. Australopithecus

you cant factor out e^(-x^2)

234. anonymous

$xe^{x^2}(\lim_{h \rightarrow 0} \frac{e^{-(h+x)^2} - e^{x^2}}{h})$

235. Australopithecus

e^(x^2)e^(x^2) =1?

236. Australopithecus

$xe^x (\lim_{h \rightarrow 0} \frac{e^{-(x + h)^2- e^{-x^2}}}{h})$

237. Australopithecus

oops lol but yeah

238. Australopithecus

that term shouldnt be in the exponential but yeah

239. anonymous

so I am thiking about these rules.. $e^{a}e^{-a} = e^{a-a} = e^{0} = 1$ and how they relate to $xe^{x^2} (\lim_{h \rightarrow 0} \frac{e^{-(x + h)^2}- e^{-x^2}}{h})$ btw.. is that function correct?

240. Australopithecus

yes

241. Australopithecus

That is it :D

242. Australopithecus

now what if f(x) considering that is the definition of a derivative

243. Australopithecus

is* not if

244. Australopithecus

you almost had it before but you there was a mistake

245. anonymous

Im assuming it is supposed to be what we started with? or is it going to be something different?

246. Australopithecus

it is something different

247. Australopithecus

when we are done you have to plug this back into the original problem

248. Australopithecus

then you will have the derivative

249. anonymous

xe^{x^2} ? the left portion of that equation ?

250. Australopithecus

We just isolate the limit we are dealing with because the other stuff would just be a waste to write over and over

251. Australopithecus

do you recognize that the limit we have now is in the form: $\frac{f(x+h) + f(x)}{h}$

252. anonymous

yes

253. Australopithecus

then what is f(x)?

254. anonymous

ah ok.. lol

255. anonymous

$e^{-x^2}$

256. Australopithecus

yes

257. Australopithecus

so the solution to that limit is just the derivative of e^(-x^2)

258. Australopithecus

do you mind just accepting that, the derivative of: $e^{f(x)} = f'(x)e^{f(x)}$ This is due to a thing called chain rule

259. Australopithecus
260. anonymous

I am ready to accept such things

261. Australopithecus

lol

262. Australopithecus

ok

263. anonymous

I dont know how to thank you enough.. this was awesome

264. Australopithecus

so then you just need to use the definition of the derivative to find the derivative of -x^2 Let us say f(x) = x^(-2) thus: $xe^{x^2}( \lim_{h \rightarrow 0} \frac{e^{-(x+h)^2}-e^{-x^2}}{h}) = xe^{x^2}(f'(x)e^{f(x)})$

265. Australopithecus

sorry lets say f(x) = -x^(2)

266. Australopithecus

that negative is important almost forgot it

267. Australopithecus

then you plug that back into your main equation when you find f'(x)

268. Australopithecus

Do some simplifying and you are done

269. Australopithecus

sorry I made a mistake in writing above, $\frac{d}{dx} e^{f(x)} = f'(x)e^{f(x)}$

270. Australopithecus

this is what you have to accept this property is due to chain rule

271. anonymous

okay cool, that seems intuitive enough

272. Australopithecus

273. anonymous

well I couldnt teach it .. but I could read it over again, and I think I'd be pretty good to this point.

274. Australopithecus

Could you do me the favour of rating me as a qualified helper? I will be around tomorrow if you need help with any of the algebra

275. Australopithecus

this question is weird

276. Australopithecus

so its ok if you cant teach it as long as you learned stuff along the way

277. anonymous

I know that mathematica returns a function $f`[x] = e^{-x^2} -2x^2 e^{-x^2}$ for $f[x] = x/e^{x^2}$ and this has bridged a big gap in working out how it came up with that equation

278. Australopithecus

well yeah you have e^(-x^2) in your equation

279. anonymous

totally on high Quality help ..

280. Australopithecus

the second part of the equation is that limit we have been working on

281. Australopithecus

you can rate me by hitting the orange button at the top

282. Australopithecus

But yeah, to write out

283. anonymous

argh.. damnit.. I clicked the wrong icon.. ah crap

284. Australopithecus

what you have fully now

285. anonymous

doesnt have undo

286. anonymous

ah man, sorry, you really deserve this one x 100

287. Australopithecus

$e^{-x^{2}}(xe^{x^2}(\lim_{h \rightarrow 0} \frac{e^{-(x+h)^2}-e^{-x^2}}{h}) + 1)$

288. Australopithecus

oh you rated someone else?

289. anonymous

yeah by accident.. hit the wrong icon

290. Australopithecus

@Preetha can you allow a redo on rating?

291. Australopithecus

not a big deal

292. Australopithecus

but you see how this all works out to give you the derivative

293. Australopithecus

she probably gets over a billion notifications

294. Australopithecus

well anyways I hope I was helpful

295. anonymous

so are you saying that this equation we arrived at , can be applied to all functions?

296. Australopithecus

no just in this case you can do this to solve this problem

297. Australopithecus

I would recommend learning the rules for doing derivatives it is much faster, not that the definition of the derivative is not useful in some case

298. anonymous

well it helped me learn some things.. but yeah.. using the rules is definitely the go

299. hartnn

just as an alternative approach for $$\large \lim \limits_{h \rightarrow 0} \frac{x(e^{-2hx-h^2}-1)}{h}$$ $$\large x\lim \limits_{h \rightarrow 0} \dfrac{(e^{-h(2x-h)}-1)}{h}\times \dfrac{-2x-h}{-2x-h}$$ $$\large x \lim \limits_{h(-2x-h) \rightarrow 0} \dfrac{(e^{-h(2x-h)}-1)}{h(-2x-h)} \lim \limits_{h \rightarrow 0}(-2x-h)$$ $$\large x[1\times (-2x-0)])$$ = -2x :)

300. Australopithecus

you made a mistake it should be -2x^2

301. Australopithecus

he forgot about the x he pulled out of the limit

302. anonymous

ah right on yeah..

303. anonymous

was this at the stage we had to still add back in $e^{-x^2}$ ?

304. Australopithecus

I dont really follow his method

305. Australopithecus

i Just know he made a mistake

306. Australopithecus

just follow what I wrote and you will solve this problem anyways i am going to sleep now goodnight

307. anonymous

well thank you very much.. and Im doing the same

308. hartnn

yes, i forgot the x that i pulled outside, thanks! it'll be -2x^2 and yes again, e^(-x^2) was pulled out earlier.

309. anonymous

and I think there was a +1 in the equation as well that we eliminated around that stage.. so we have here now, all the pieces to lead us from $f[x] = x/e^{x^2}$ to $f'[x]= e^{-x^2}(1-2x^2 )$ $f'[x]= e^{-x^2}-2x^2 e^{-x^2}$ Thank you guys.