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Question
\[\frac{x^2 - 1}{x}\]
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{(x + h )^2 - 1}{x + h} - \frac{x^2 - 1}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x^2 + 2xh + h^2 - 1}{x + h} - \frac{x^2 - 1}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{x^3 + 2 x^2 h + h^2 x - x - x^3 - x^2 h + x + h}{xh(x + h)}\]
\[ = \lim_{h \to 0} \frac{x^2 h + h^2 x + h}{x(x + h)}\]
\[ = \lim_{h \to 0} \frac{h( x^2 + hx + 1)}{xh(x + h)}\]
\[ = \lim_{h \to 0} \frac{x^2 + hx + 1}{x(x + h)}\]
\[ = \frac{x^2 + 1}{x^2}\]
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