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