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Differentiate (x^2)^(ln[x])

  1. May 14, 2009 #1
    1. The problem statement, all variables and given/known data

    differentiate (x2)lnx


    im having a blonde moment...how do you start?
     
  2. jcsd
  3. May 14, 2009 #2

    Cyosis

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    Write x as [tex]e^{\ln x}[/tex].
     
  4. May 14, 2009 #3
    ok so you mean write

    eln(x2)^ln(x)

    this gives me eln(x)*ln(x2)
     
  5. May 14, 2009 #4

    Cyosis

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    Yes that's correct. You can write the part in the exponent slightly easier by using [itex]\ln x^2=2\ln x[/itex]. Now you just have to use the chain rule.
     
    Last edited: May 14, 2009
  6. May 14, 2009 #5
    how can you say this?
     
  7. May 14, 2009 #6

    Cyosis

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    I 'said' that, because in the latex code I wrote 2\lnx, \lnx is not a command in latex so it doesn't recognize it. I added the space now so it becomes 2\ln x, which should display the correct result.

    Edit: I see you edited post 3, but what you did there is not correct. [tex](x^2)^{\ln x}=(x)^{2 \ln x}=(e^{\ln x})^{\ln x^2}=e^{\ln x \ln x^2} \neq e^{(\ln x^2)^{\ln x}}[/tex].
     
    Last edited: May 14, 2009
  8. May 14, 2009 #7

    HallsofIvy

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    He didn't say it. (Unless he editted his post immediately after your response.) He said ln(x2)= 2ln(x).

    On edit: And while I was typing this, he explained!

    Using that
    [tex](x^2)^{ln(x)}= e^{(ln(x))(2ln(x))}= e^{2(ln(x))^2}[/itex]
     
  9. May 14, 2009 #8
    ok i got

    e(2(lnx)2)

    differentiating gives

    (4lnx)/x * e(2(lnx)2)
     
  10. May 14, 2009 #9

    HallsofIvy

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    Yes, that's right.

    Another way to differentiate [itex]y= (x^2)^{ln x}[/itex] is to take the logarithm of both sides: [itex]ln(y)= ln((x^2)^{ln(x)}= ln(x)(ln(x)^2)= 2 (ln(x))^2[/itex]

    Now, differentiating both sides with respect to x,
    [tex]\frac{1}{y}y'= 4 ln(x)\frac{1}{x}[/tex]
    [tex]y'= \frac{4 ln(x)}{x} y= \frac{4 ln(x)}{x}(x^2)^{ln(x)}[/tex]

    (If you were given the problem as [itex](x^2)^{ln x}[/itex] it is probably better to write the answer using that rather than [itex]e^{2(ln x)^2}[/itex].)
     
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