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Math Help - analysis I

  1. #1
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    analysis I

    suppose that f is differentiable at a and f(a) is not equal to 0.
    show that for h sufficiently small f(a+h) is not equal to zero and using the definition of a derivative directly, prove that 1/(f(x)) is differentiable at x=a and
    (1/f)'(a) = -f'(a)/f^2(a)

    stuck on this one anyone have any ideas? is the second part just plug and chug?
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  2. #2
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    Quote Originally Posted by gap135 View Post
    suppose that f is differentiable at a and f(a) is not equal to 0.
    Do you mean that f'(a) \not= 0?
    Otherwise, because f is differentiable at a then f is continuous at a.
    If f is not zero at a then there is a neighborhood of a throughout which f is not zero.
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  3. #3
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    re

    It's f(a) not f'(a)
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  4. #4
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    Then the first part is trival. If f is contiouous at a and f(a) is not zero then if (x) is not zero on some neighborhood of a.
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  5. #5
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    Quote Originally Posted by gap135 View Post
    , prove that 1/(f(x)) is differentiable at x=a and
    (1/f)'(a) = -f'(a)/f^2(a)
    ?
    What does it means (1/f)'(a)?
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