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