1. ## 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

2. $
\begin{gathered}
x^2 a - 3x^2 + a - a^2 x + 3a^2 - x = \hfill \\
= ax\left( {x - a} \right) + a - 3x^2 + 3a^2 - x = \hfill \\
= ax\left( {x - a} \right) - (x - a) - 3\left( {x^2 - a^2 } \right) = \hfill \\
\end{gathered}
$

$\begin{gathered}
= \left( {ax - 1} \right)\left( {x - a} \right) - 3\left( {x - a} \right)\left( {x + a} \right) = \hfill \\
= \left( {ax - 1 - 3x - 3a} \right)\left( {x - a} \right) \hfill \\
\end{gathered}$