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Thread: finding f`(a)

  1. #1
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    finding f`(a)

    I am having trouble with the following problem.

    Let f(x) = cubedroot of x
    if a is not equal to 0, use f`(a) = lim as x approaches a {f(x)-f(a)}/(x-a) to find f`(a)

    I put it as lim as h approaches 0 (cubed root (a+h)-cubed root (a))/h but am unsure if this is correct and what I should do next.

    Thanks
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  2. #2
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    Let $\displaystyle x^{\frac{1}{3}} = y$ and $\displaystyle a^{\frac{1}{3}} = b$, then:

    $\displaystyle \displaystyle \lim_{x\to{a}}\frac{x^{\frac{1}{3}}-a^{\frac{1}{3}}}{x-a} = \lim_{y\to{b}}\frac{y-b}{y^3-b^3} = \lim_{y\to{b}}\frac{1}{b^2+b y+y^2} = \frac{1}{3b^2}.$

    Since $\displaystyle b = a^{\frac{1}{3}}$, we have:

    $\displaystyle \displaystyle f'(a) = \frac{1}{3a^{\frac{2}{3}}}.$
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  3. #3
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    Thanks a lot m8. That was great.
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