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 3 years ago
I am stuck with an easy problem  I must be missing something. I am trying to use the limit process to find the derivative of f(x) = 8 / sqrt of x. Can you help me see where I am messing up?
 3 years ago
I am stuck with an easy problem  I must be missing something. I am trying to use the limit process to find the derivative of f(x) = 8 / sqrt of x. Can you help me see where I am messing up?

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FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0\[f(x) = 8/\sqrt{x}\] find the derivative using the limit process

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0f'(x) = lim \[f^\prime(x) =\lim_{\Delta x \rightarrow 0} [(8/\sqrt{x+c} 8/\sqrt{x})/c]\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I am banging my head against this problem and getting a bad answer every time. We can tell by looking at it that the answer is\[(4/x^(3/2))\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Well, I think you see what I mean...

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0But I haven't figured out the algebra to leave me with a 4 in the numerator. I keep getting stuck with things that don't work out in the end.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Weeku, what do you have for me so far?

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Whoops, I made a mistake. Darn editor. . . Hold on.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Seems no matter what I try, I end up with delta x /[( sqrt x + delta x) + sqrt x]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I am going in circles.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0are you still there?

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Yeah. I'm stuck as well. Hrm. . . I'm still trying, though. Sorry xD Interesting problem, though.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Ah. Got it. Took a whole looseleaf paper to work out. . . Pretty nasty.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Did you get up to \[(8\sqrt{x}  8\sqrt{x+h}) / (h \sqrt{x}\sqrt{x+h})\]? Or, do you know how I got there? At that point, multiple the numerator and denominator by the conjugate of the numerator.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Then it becomes a lot of distributing and factoring, until you can finally divide out the h. Once that happens, substitute 0 for h. Phew. I dunno if I can type out everything I did. . . I probably did it the long way, but I saw no other, shorter way. Do you want me to type it out?

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Yes, I got that far. I see you have h = delta x and you canceled out one h already.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I would be grateful if you typed it out.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Before that, all I did was find the common denominator of 8/sqrt[x+h] and 8/sqrt[x], add them together, and instead of dividing by h, multiply by 1/h to get \[8\sqrt{x}  8\sqrt{x+h} / h \sqrt{x} \sqrt{x+h}\] Then I factored out the 8 to get the above ^ Oh. Okay.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Whoops. Never mind about factoring out the 8.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1I tried doing that, with no luck. :D

Ishaan94
 3 years ago
Best ResponseYou've already chosen the best response.3\[\frac{\frac{8}{\sqrt{x+h}}  \frac{8}{\sqrt{x}}}{h}\]Firs Only the Numerator.... \[8\frac{\sqrt{x}  \sqrt{x+h}}{\sqrt{x+h}\times \sqrt{x}}\] Rationalization of Numerator \[\frac{8}{\sqrt{x+h}\times\sqrt{x}\times(\sqrt{x} + \sqrt{x+h})}\] \[\frac{8}{2\times x^{\frac{3}{2}}}\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Ishaan94, I am looking over what you have done to see where I diverged. Thank you.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1I lost you after "Rationalization of Numerator," but nicely done. I did it the very, very long way, I see. . . Sorry about that, FreeTrader.

Ishaan94
 3 years ago
Best ResponseYou've already chosen the best response.3\[\Huge{\frac{\frac{8\cancel{h}}{\sqrt{x+h}\times\sqrt{x}\times(\sqrt{x} + \sqrt{x+h})}}{\cancel{h}}}\]

Ishaan94
 3 years ago
Best ResponseYou've already chosen the best response.3\[8\frac{\sqrt{x}  \sqrt{x+h}}{\sqrt{x} \times \sqrt{x+h}} \times \frac{\sqrt{x} + \sqrt{x+h}}{\sqrt{x} + \sqrt{x+h}}\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I see how Ishaan94 got to the "first only the numerator..." expression. But I am lost on how he got rid of the terms in the numerator.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Right, multiply by conjugate.

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0I really think you are going about this the wrong way. You should change the square root into a power (1/2) and then bring the denominator to the top by changing the power to negative. Then you can simply use the limit process of: f(x)= 8x^(1/2) f(x+h)=8(x+h)^(1/2) And plug this into the derivative formula. You should get rid of the terms in the denominator.

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0Then factor out a h and cancel.

Ishaan94
 3 years ago
Best ResponseYou've already chosen the best response.3\[8\frac{x  (x+h)}{\sqrt{x}\times \sqrt{x+h} \times (\sqrt{x} + \sqrt{x+h})}\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I think it's dawning on me that I missed distributing the 1 on the h, too.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Stick with me, I am going to check that on my paper.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Ishaan94, when I perform the operation 8 *[ x (x+h) ] I am left with 8x 8(x+h) which gives 8x 8x 8h

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Which gives 8h. divide out the h's.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I am onto this now... working the algebra. searching for the lightswitch for the bulb over my head...

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0\[f(x)=\frac{8}{\sqrt{x}}\] \[f(x)=\frac{8}{x ^{1/2}}\] \[f(x)=8x ^{\frac{1}{2}}\] \[f(x+h)=8(x+h)^{1/2}\] \[f \prime(x)\lim_{h \rightarrow 0}=I can't remember the formula\] Now just use you derivative formula, cancel where necisary, factor out a h, then cancel with the denominator.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0My problem is now in the denominator of the numerator. I don't see a h to cancel the one against the 8.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1You're forgetting that the entire equation is over h at this point. \[\Huge{\frac{\frac{8\cancel{h}}{\sqrt{x+h}\times\sqrt{x}\times(\sqrt{x} + \sqrt{x+h})}}{\cancel{h}}}\]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I have that now. Thanks. I am down to working with the remaining denom terms.

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0All terms on the right hand side of the minus sign in your derivative formula that have a x term should cancel with the terms. I think you're going about this the wrong way. Instead of having a quotient you should really bring the denominator to the top by making the power negative. Remember this rule? am^x = a/m^x

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Yes. I am already so deep in the prob that I am going to finish it with frax.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Here's what I have in the denominator, which I am tempted to misspell as demonator. \[\sqrt{x} * \sqrt{x} * \sqrt{x+h} + \sqrt{x} * \sqrt{x+h} * \sqrt{x+h} \]

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Since the h values go to zero in the limit, the following is the denominator in the limit: \[2\sqrt{x}\sqrt{x}\sqrt{x}\]

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0\[f(x)=\frac{8}{\sqrt{x}}\] \[f(x)=\frac{8}{x ^{1/2}}\] \[f(x)=8x ^{\frac{1}{2}}\] \[f(x+h)=8(x+h)^{1/2}\] \[\lim_{h \rightarrow 0}f'(x)=\frac{f(x+h)f(x)}{h}\] lim{h>0) of f'(x) = (8(x+h)^(1/2)  8x^(1/2))/h You can do the rest, I'm sure.

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Yes, freetrader, (sqrt[x])^3 = x^(3/2)

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0Okay, I am losing my sanity. Can you help me remember if \[\sqrt{x^3} = \sqrt{x}\sqrt{x}\sqrt{x}\] ???

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0Yes  and this should be in the denominator of the equation once it has been solved. Notice how the ^(1/2) becomes 1.5 when you increase the power by one. And yes it does FreeTrader

Weeku
 3 years ago
Best ResponseYou've already chosen the best response.1Yup. That is a true statement.

chaise
 3 years ago
Best ResponseYou've already chosen the best response.0Stop writing it as a square root until you get to the final stage of the question. sqrt(x)^3 = x^(3/2)

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0I have temp insanity.

FreeTrader
 3 years ago
Best ResponseYou've already chosen the best response.0So, my answer took me 7 hours to reach.
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