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Thread: Finding the derivative

  1. #1
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    Finding the derivative

    f(x) = 1/x
    a = 2

    Use the definition:
    f' (a) = lim as x approaches a of f(x) - f(a) / x - a

    Can someone help?
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  2. #2
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    Quote Originally Posted by Morgan82 View Post
    f(x) = 1/x
    a = 2

    Use the definition:
    f' (a) = lim as x approaches a of f(x) - f(a) / x - a

    Can someone help?
    $\displaystyle f(x) = \frac{1}{x}$

    $\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

    $\displaystyle = \lim_{h \to 0}\frac{\frac{1}{x + h} - \frac{1}{x}}{h}$

    $\displaystyle = \lim_{h \to 0}\frac{\frac{x - (x + h)}{x(x + h)}}{h}$

    $\displaystyle = \lim_{h \to 0}\frac{\frac{-h}{x(x + h)}}{h}$

    $\displaystyle = \lim_{h \to 0}\left(\frac{-h}{x(x + h)}\cdot\frac{1}{h}\right)$

    $\displaystyle = \lim_{h \to 0}\frac{-1}{x(x + h)}$

    $\displaystyle = -\frac{1}{x^2}$.


    Therefore

    $\displaystyle f'(a) = -\frac{1}{a^2}$.
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