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Homework Help: Derivative of y = x^(x^2-7)

  1. Sep 5, 2006 #1
    i jst wanted to know if this is right. I need to find out the derivative of:
    y = x^(x^2 - 7) :smile:

    y = x^(x^2 - 7)

    ln y = (x^2 -7) ln (x)

    1/y = x^2 + 2x -7
    x

    y = x
    (x^2 + 2x - 7)

    dy/dx = -x^2 + 2x - 9 <----------- my answer
    (x^2 + 2x -7)
     
  2. jcsd
  3. Sep 5, 2006 #2

    TD

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    Homework Helper

    Your notation is a bit unclear to me, but the logarithm is a good idea:

    [tex]
    y = x^{x^2 - 7} \Rightarrow y = \exp \left( {\ln \left( {x^{x^2 - 7} } \right)} \right) = \exp \left( {\left( {x^2 - 7} \right)\ln \left( x \right)} \right)
    [/tex]

    Can you find the derivative of that exponential?
     
  4. Sep 5, 2006 #3

    HallsofIvy

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    Science Advisor

    Beware of non-fixed fonts! Much better in Latex.
    I don't see how you got from x2- 7 to x2+ 2x- 7 and I certainly don't see where that -x2+ 2x- 9 came from!
    And, you seem to first solve for y, then it magically becomes y'.

    You want to differentiate [itex]y= x^{x^2-7}[/itex] so you rewrite it as
    [tex]ln(y)= (x^2- 7)ln(x)[/itex]
    The derivative of ln y with respect to y is [itex]\frac{1}{y}[/itex] but you want the derivative with respect to x- so use the chain rule:
    [tex]\frac{d ln y}{dx}= \frac{d ln y}{dy}\frac{dy}{dx}= \frac{1}{y}\frac{dy}{dx}[/tex]
    but it is exactly dy/dx you want to find!
    The left hand side is not just 1/y but is (1/y)y'.

    On the right side you want to differentiate (x2- 7)ln x: use the product rule- (fg)'= f'g+ fg'. ((x2-7)ln(x))'= (x2-7)' ln(x)+ (x2-7)(ln x)'. The derivative of x2- 7 is 2x and the derivative of ln x is 1/x so ((x2-7)ln(x))'= (2x) ln(x)+ (x2-7)/x. Put those together:
    [tex]\frac{1}{y}y'= 2x ln x+ \frac{x^2- 7}{x}[/tex]
    and solve for y'.
     
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