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Differentiation problem

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data
    This is the larger problem to the small portion that I already posted in the Precalc Hw help forum. I still can't figure out how to get to the answer.

    The problem is this:
    I am trying to find the derivative of [tex]f(x)=x+\frac{9}{x}[/tex].

    2. Relevant equations
    I know via power rule that the answer will be:
    [tex]\frac{dy}{dx}=1+\frac{9}{x^{2}}[/tex]
    However, I must do it the messy way. :grumpy:

    3. The attempt at a solution
    This is what I have got so far:
    1.[tex]\frac{f(x+h)-f(x)}{h}[/tex]

    2.[tex]=\lim_{h\rightarrow0}\frac{\left((x+h)+\frac{9}{x+h}\right)-\left(x+\frac{9}{x}\right)}{h}[/tex]

    3.[tex]=\lim_{h\rightarrow0}\frac{x+h+\frac{9}{x+h}-x-\frac{9}{x}}{h}[/tex]

    4.[tex]=\lim_{h\rightarrow0}\frac{h}{h}+\frac{\frac{9}{x+h}}{h}-\frac{\frac{9}{x}}{h}[/tex]

    5.[tex]=\lim_{h\rightarrow0}1+\left(\frac{9h}{x+h}-\frac{9h}{x}\right)[/tex]

    6.[tex]=\lim_{h\rightarrow0}1+\left(\frac{x}{x}\cdot\frac{9h}{x+h}-\frac{9h}{x}\cdot\frac{x+h}{x+h}\right)[/tex]

    7.[tex]=\lim_{h\rightarrow0}1+\frac{9hx-9hx-9h^{2}}{x(x+h)}[/tex]

    8.[tex]=\lim_{h\rightarrow0}1+\frac{-9h^{2}}{x(x+h)}[/tex]

    I don't know where to go from here or even if here is where I am supposed to be. Help please?
     
    Last edited: Sep 14, 2009
  2. jcsd
  3. Sep 14, 2009 #2
    Partial whos-a-what? :confused:
     
  4. Sep 14, 2009 #3

    Dick

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    Science Advisor
    Homework Helper

    You were doing fine until step 5. Then you changed (9/(x+h)-9/x)/h into (9h/(x+h)-9h/x). You can't do that. You just moved h into the numerator by 'magic'. Leave it in the denominator! You also don't want to multiply by h/h in the next step. You want to multiply by x/x to get the common denominator.
     
  5. Sep 14, 2009 #4
    Ok thats where I went wrong. Thanks for your help. :biggrin: Btw the multiplying h/h was a typo, its supposed to be x/x.
     
  6. Sep 14, 2009 #5
    Awesome. So going by what you said Dick, I figured out that I messed up by putting that h in the numerator (slopy fraction solving :) ) I worked it out and got the right answer. Thanks for your help Dick.
     
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