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