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amy0799

  • one year ago

if f(x) = 4/(x-2), find f'(x)

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  1. anonymous
    • one year ago
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    i am guessing that this is the very beginning of calc and you have to do with by hand, not using any shortcuts am i right?

  2. amy0799
    • one year ago
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    yup

  3. anonymous
    • one year ago
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    too bad

  4. anonymous
    • one year ago
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    next week you will say \[f'(x)=-\frac{4}{(x-2)^2}\] in your head, but i guess we can take the steps (long and arduous steps)

  5. anonymous
    • one year ago
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    we will have to go nice an slow first write the definition, then use some algebra ok a lot of algebra it is 99% algebra

  6. anonymous
    • one year ago
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    did you get to \[\huge \lim_{h\to 0}\frac{\frac{4}{(x+h-2)}-\frac{4}{x-2}}{h}\]

  7. amy0799
    • one year ago
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    yes.

  8. anonymous
    • one year ago
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    ok now in order to spare some agony lets forget about the h in the denominator, and forget about the limit we will deal with those last, ok?

  9. amy0799
    • one year ago
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    ok

  10. anonymous
    • one year ago
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    so the algebra we need is to (carefully) do this subraction \[\frac{4}{x+h-2}-\frac{4}{x-2}\]

  11. amy0799
    • one year ago
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    (4(x-2)-4(x+h-2))/((x-2)(x+h-2) is that right?

  12. anonymous
    • one year ago
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    can you do this? leave the denominator in factored form i.e don't multiply out

  13. anonymous
    • one year ago
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    yes that is correct now carefully multiply out in the numerator and combine like terms

  14. amy0799
    • one year ago
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    (4x-4x-8+8-4h)/((x-2)(x+h-2))

  15. anonymous
    • one year ago
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    ok that is multiplying out what is left in the numerator?

  16. amy0799
    • one year ago
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    -4h

  17. anonymous
    • one year ago
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    exactly!

  18. anonymous
    • one year ago
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    now recall there is an \(h\) in the denominator cancel it

  19. anonymous
    • one year ago
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    what is left?

  20. amy0799
    • one year ago
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    -4/((x-2)(x+h-2))

  21. anonymous
    • one year ago
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    bingo

  22. anonymous
    • one year ago
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    now 'take the limit as h goes to zero" which is a fancy way of saying erase that h what is left?

  23. amy0799
    • one year ago
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    -4/(x-2)^2

  24. anonymous
    • one year ago
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    as promised (see above)

  25. amy0799
    • one year ago
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    thank u so much! :D

  26. anonymous
    • one year ago
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    YW!

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