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Math Help - Explain Derviations of potensfunctions in laymens terms please

  1. #1
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    Explain Derviations of potensfunctions in laymens terms please

    Okay sitting here doing some homework that is using the simple derivaiton of potensfunctions... and I can't get my math to work here.
    Please explain how you solve these one;

    y = 6th square of x
    y`= ?

    Any help here is appreciated!
    Thanks in advance
    Last edited by Hanga; April 16th 2008 at 10:01 AM.
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  2. #2
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    Quote Originally Posted by Hanga View Post
    Okay sitting here doing some homework that is using the simple derivaiton of potensfunctions... and I can't get my math to work here.
    Please explain how you solve these one;

    y = 6th square of x
    y`= ?

    Any help here is appreciated!
    Thanks in advance

    6th square of x??
    Do you mean \sqrt[6] {x}?
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  3. #3
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    Quote Originally Posted by Isomorphism View Post
    6th square of x??
    Do you mean \sqrt[6] {x}?
    Yeah indeed I do
    I dunno how the Math commands on these forums work :P

    anyways

    whats the derivation of y = \sqrt[6] {x}
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by Hanga View Post
    Yeah indeed I do
    I dunno how the Math commands on these forums work :P

    anyways

    whats the derivation of y = \sqrt[6] {x}
    y = \sqrt[6] {x}

    y = {x}^{\frac16}

    y' = \frac16{x}^{\frac16 - 1}

    y' = \frac16{x}^{-\frac56}

    y' = \frac16 \sqrt[6]{\frac1{x^5}}
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  5. #5
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    Hello, Hanga!

    I don't think a "layman" could handle this one . . .


    Differentiate: . f(x) \:=\:x^{\frac{1}{6}} . . . . using the definition of a derivative (?)

    We have: . f(x+h) - f(x) \;= \;\frac{(x+h)^{\frac{1}{6}} - x^{\frac{1}{6}}}{1}

    Now we do something bizarre . . . multiply top and bottom by:
    . . (x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}

    and we get: . \frac{(x + h) - x} {(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}}

    . . . . . . . = \;\frac{h}{(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}}


    \text{Divide by }h\!:\;\;\frac{1}{(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}}


    Take the limit:

    \lim_{h\to0}\left[\frac{1}{(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}} \right]

    . . = \;\frac{1}{x^{\frac{5}{6}} + x^{\frac{4}{6}}x^{\frac{1}{6}} + x^{\frac{3}{6}}x^{\frac{2}{6}} + x^{\frac{2}{6}}x^{\frac{3}{6}} + x^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}} . = \;\frac{1}{x^{\frac{5}{6}} + x^{\frac{5}{6}} + x^{\frac{5}{6}} + x^{\frac{5}{6}} + x^{\frac{5}{6}} + x^{\frac{5}{6}}}


    Therefore: . \boxed{f'(x) \;=\;\frac{1}{6x^{\frac{5}{6}}} }

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  6. #6
    Moo
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    Hello,

    Why getting so complicated ? o.O
    I think that people who still don't know general formulaes of derivation won't know they have to multiply by such a thing
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  7. #7
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    Quote Originally Posted by Moo View Post
    Hello,

    Why getting so complicated ? o.O
    I think that people who still don't know general formulaes of derivation won't know they have to multiply by such a thing
    Soroban did it using the definition of the derivative, not the power rule

    (Okay now I'm seriously going to bed )
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  8. #8
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    Quote Originally Posted by Soroban View Post

    . . . . . . . = \;\frac{h}{(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}}


    \text{Divide by }h\!:\;\;\frac{1}{(x+h)^{\frac{5}{6}} + (x+h)^{\frac{4}{6}}x^{\frac{1}{6}} + (x+h)^{\frac{3}{6}}x^{\frac{2}{6}} + (x+h)^{\frac{2}{6}}x^{\frac{3}{6}} + (x+h)^{\frac{1}{6}}x^{\frac{4}{6}} + x^{\frac{5}{6}}}


    Am I missing something? You divided h from the top but not from the bottom?
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  9. #9
    Moo
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    He divided by h : f(x+h)-f(x)=h/...

    By taking the limit -> derivative
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  10. #10
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    Ooh! He didn't divide by h yet. I see I see.
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  11. #11
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    Thanks guys!

    Not only did you show me the correct answer, you also made me see how to prove that the calcutlation is correct.. hallelujah!
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