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Math Help - Formal Definition of Derivative

  1. #1
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    Formal Definition of Derivative

    I need to use the formal definition of a derivative,
    g'(x) = lim h->0 (f(x + h) - f(x)) / h

    to solve x^(2/3)...

    lim h->0 (f(x+h)^(2/3) - x^(2/3)) / h

    I know the answer is obviously (2/3)x^(-1/3) but we're supposed to use the difference quotient as stated above. Any help would be greatly appreciated
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by DoQrs View Post
    I need to use the formal definition of a derivative,
    g'(x) = lim h->0 (f(x + h) - f(x)) / h

    to solve x^(2/3)...

    lim h->0 (f(x+h)^(2/3) - x^(2/3)) / h

    I know the answer is obviously (2/3)x^(-1/3) but we're supposed to use the difference quotient as stated above. Any help would be greatly appreciated
    g(x) = x^{2/3}

    \Rightarrow g'(x) = \lim_{h \to 0} \frac {g(x + h) - g(x)}h

    = \lim_{h \to 0} \frac {(x + h)^{2/3} - x^{2/3}}h

    now simplify the top and find the limit. i believe rationalizing the top (multiplying by its conjugate over itself) should work
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  3. #3
    MHF Contributor red_dog's Avatar
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    \displaystyle g'(x)=\lim_{h\to 0}\frac{\sqrt[3]{(x+h)^2}-\sqrt[3]{x^2}}{h}=
    \displaystyle=\lim_{h\to 0}\frac{(x+h)^2-x^2}{h\left(\sqrt[3]{(x+h)^4}+\sqrt[3]{x^2(x+h)^2}+\sqrt[3]{x^4}\right)}=
    \displaystyle=\lim_{h\to 0}\frac{h(2x+h)}{h\left(\sqrt[3]{(x+h)^4}+\sqrt[3]{x^2(x+h)^2}+\sqrt[3]{x^4}\right)}=
    \displaystyle=\frac{2x}{3\sqrt[3]{x^4}}=\frac{2}{3\sqrt[3]{x}}=\frac{2}{3}x^{-\frac{1}{3}}
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    Thank you, did you use an online tool to generate the images?
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  5. #5
    MHF Contributor red_dog's Avatar
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    We use Latex.
    See "Latex Help" category, in the home page on this forum.
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by DoQrs View Post
    Thank you, did you use an online tool to generate the images?
    or see our LaTex tutorial here
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