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iop360

  • 2 years ago

Let y = (7/x^3)(√196x^6 +364x^2 + (169/x^2)) Find the derivative of y. Hint: Simplify y before computing the derivative. Answer: (-364/x^5)

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  1. iop360
    • 2 years ago
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    does anyone know how to simplify the equation first before taking it's derivative?

  2. ChmE
    • 2 years ago
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    can you rewrite the equation using the equation tool on this site. It is tough to see in this form.

  3. iop360
    • 2 years ago
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    yep

  4. iop360
    • 2 years ago
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    \[y = \frac{ 7 }{ x^3 }\sqrt{196x^6 + 364x^2 + \frac{ 169 }{ x^2 }}\]

  5. iop360
    • 2 years ago
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    Answer is \[\frac{ dy }{ dx } = \frac{ -364 }{ x^5 }\]

  6. ChmE
    • 2 years ago
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    I don't know how to simplify the radical. I would only know how to differentiate right away and it'll get messy

  7. iop360
    • 2 years ago
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    ok thanks anyway

  8. iop360
    • 2 years ago
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    you would use the product rule right?

  9. ChmE
    • 2 years ago
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    ya.

  10. ChmE
    • 2 years ago
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    I'm doing it too. it's getting pretty rough

  11. iop360
    • 2 years ago
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    \[y' = -21x ^{-4} (196x^6 + 364x^2 + 169x^{-2})^{1/2} + 7x ^{-3}(\frac{ 1 }{ 2 } (196x^6 + 364 + 169x ^{-2})^{-1/2}(1176x^5 +728x -338x ^{-3})^{-1/2}\]

  12. iop360
    • 2 years ago
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    whoops lol

  13. iop360
    • 2 years ago
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    simplifying is the hardest part

  14. zepdrix
    • 2 years ago
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    Hmm there is something going on inside this square root, I'm trying to figure it out. Not quite there yet, but I think it's not long off. 196 = 14^2 169=13^2 Hmmmm

  15. zepdrix
    • 2 years ago
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    OH! yes found it! :)

  16. zepdrix
    • 2 years ago
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    364 = 2 (13*14)

  17. zepdrix
    • 2 years ago
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    (14x^4 + 13)^2 I think it'll simplify like that.

  18. ChmE
    • 2 years ago
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    nice

  19. iop360
    • 2 years ago
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    so how would this look like

  20. iop360
    • 2 years ago
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    after simplifying?

  21. zepdrix
    • 2 years ago
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    |dw:1348968835510:dw| Hmm, I think it simplifies like this, looks much easier to deal with :) I got lazy on the multiplying there hehe

  22. iop360
    • 2 years ago
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    thanks

  23. ChmE
    • 2 years ago
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    I got the same thing. great job finding the factorization pattern @zepdrix

  24. iop360
    • 2 years ago
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    how it go from the original to this though

  25. ChmE
    • 2 years ago
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    Ok I'll type this one out

  26. ChmE
    • 2 years ago
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    I'm going to leave off the radical for now ok?

  27. iop360
    • 2 years ago
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    k

  28. ChmE
    • 2 years ago
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    \[196x ^{6}+363x ^{2}+\frac{ 169 }{ x ^{2} }\]Find a common denominator and add\[\frac{ 196x ^{8}+364x ^{4}+169 }{ x ^{2} }\]rewrite the constants\[\frac{ 14*14x ^{8}+2*14*13x ^{4}+13*13 }{ x ^{2} }\]You still with me so far

  29. ChmE
    • 2 years ago
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    notice that the numerator looks just like a polynomial

  30. iop360
    • 2 years ago
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    yep

  31. ChmE
    • 2 years ago
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    \[\frac{ (14x ^{4}+13)(14x^{4}+13) }{ x ^{2}}\]

  32. iop360
    • 2 years ago
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    oh alright

  33. ChmE
    • 2 years ago
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    \[\frac{ \sqrt{(14x^{4}+13)^{2}} }{ \sqrt{x^{2}} }\]

  34. ChmE
    • 2 years ago
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    You probably see it now

  35. iop360
    • 2 years ago
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    thanks!

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