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Math Help - diff. derivative

  1. #1
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    diff. derivative

    Def: Let f be a real-valued function defined on an interval I containing the point c, (we allow the possibilty that c is an endpoint of I) we say that f is differentiable at c (or has a derivative at c) if the limit

    lim x->c (f(x) - f(c))/(x-c)
    exists and is finite.

    use the definition above to find the derivative of each function.

    C) f(x) = 1/x for x is not equal to 0
    D) f(x) = sqrt(x) for x>0
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  2. #2
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    Quote Originally Posted by learn18 View Post
    Def: Let f be a real-valued function defined on an interval I containing the point c, (we allow the possibilty that c is an endpoint of I) we say that f is differentiable at c (or has a derivative at c) if the limit
    You should say I is an open interval. And c cannot be an endpoint. Otherwise the limit might not make sense.

    C) f(x) = 1/x for x is not equal to 0
    Let y!=0 then what is derivative at y, if it exists.

    (f(x) - f(y))/(x-y) = (1/x - 1/y)/(x-y) = (-(x-y)/xy)/(x-y) = -1/xy
    As x--> y we have,
    -1/y^2
    Thus,
    -1/x^2 is the derivative.
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