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Math Help - Normal Derivative

  1. #1
    Member
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    Normal Derivative

    Just can't seem to get this algebra down

    find f'(a)

    f(x) = \frac{x^2+1}{x-3}


    \lim_{x \to a} \frac{f(x)-f(a)}{x-a}  = \lim_{x \to a} \frac{\frac{x^2+1}{x-3} - \frac{a^2+1}{a-3}}{x-a} =  \lim_{x \to a} \frac{(x^2+1)(a-3)-(a^2+1)(x-3)}{(x-a)(x-3)(a-3)} = \lim_{x \to a}\frac{x^2a-3x^2+a-a^2x+3a^2-x}{(x-a)(x-3)(a-3)}

    and here I get stuck trying to get that (x-a) out of the denominator

    THanks
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  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    <br />
\begin{gathered}<br />
  x^2 a - 3x^2  + a - a^2 x + 3a^2  - x =  \hfill \\<br />
   = ax\left( {x - a} \right) + a - 3x^2  + 3a^2  - x =  \hfill \\<br />
   = ax\left( {x - a} \right) - (x - a) - 3\left( {x^2  - a^2 } \right) =  \hfill \\ <br />
\end{gathered} <br />

    \begin{gathered}<br />
   = \left( {ax - 1} \right)\left( {x - a} \right) - 3\left( {x - a} \right)\left( {x + a} \right) =  \hfill \\<br />
   = \left( {ax - 1 - 3x - 3a} \right)\left( {x - a} \right) \hfill \\ <br />
\end{gathered}
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