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anonymous

  • 5 years ago

could someone work out y'' for y= ln x/x^2

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  1. anonymous
    • 5 years ago
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    here is the working

  2. anonymous
    • 5 years ago
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    \[y=\frac{\ln x }{x^2} \rightarrow y'=\frac{x^2 (\ln x)'-\ln x (x^2)'}{(x^2)^2}=\frac{x^2/x-2x \ln x}{x^4}\]\[=\frac{x-2x \ln x}{x^4}=\frac{x-2\ln x}{x^3}\]

  3. anonymous
    • 5 years ago
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    i moved the x^2 to the top so i could use the product rule

  4. anonymous
    • 5 years ago
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    \[y''= \left( \frac{x-2 \ln x}{x^3} \right)'=\frac{x^3(x-2 \ln x)'-(x-2\ln x)(x^3)'}{(x^3)^2}\]\[=\frac{x^3(1-2/x)-(x-2\ln x)(3x^2)}{(x^3)^2}\]\[=\frac{x^3-2x^2-3x^3+6x^2 \ln x}{x^6}=\frac{x-2-3x+6\ln x}{x^4}\]\[=\frac{6\ln x-2(x+1)}{x^4}\]

  5. anonymous
    • 5 years ago
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    ballards, your way is equivalent, and actually, my preferred method! I thought you might have needed it using the quotient rule.

  6. anonymous
    • 5 years ago
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    our teacher said use whatever way we can get it. could you work it out that way so i can check my work?

  7. anonymous
    • 5 years ago
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  8. anonymous
    • 5 years ago
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    I'm rushing through stuff. Hope it's what you've got.

  9. anonymous
    • 5 years ago
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    here is what i got

  10. anonymous
    • 5 years ago
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  11. anonymous
    • 5 years ago
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    the second line is the first derivative

  12. anonymous
    • 5 years ago
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    Your result seems fine.

  13. anonymous
    • 5 years ago
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    You just need to simplify now your second derivative now.

  14. anonymous
    • 5 years ago
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    The result in my attachment is what you should end up with. Ignore what I typed up on this thing earlier. Working on this thing is impossible.

  15. anonymous
    • 5 years ago
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    so could you show me how to do that. i am stuck

  16. anonymous
    • 5 years ago
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    It's kind of difficult to explain much further online :( Are you having a problem with the working in my attachment, or in simplifying the algebra in your own answer?

  17. anonymous
    • 5 years ago
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    SIMPLIFYING

  18. anonymous
    • 5 years ago
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    I would first multiply each of your products to get four little expressions, and then collect like terms. I started typing but it's going to take forever. I will do it on paper and scan.

  19. anonymous
    • 5 years ago
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    okay thank you so much!

  20. anonymous
    • 5 years ago
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