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Question
Differentiate of the following from first principle:
(−x)−1
Solution
\[ \left( - x \right)^{- 1} = \frac{1}{- x} \]
\[\frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( \frac{1}{- x} \right) = \lim_{h \to 0} \frac{\frac{1}{- \left( x + h \right)} - \frac{1}{- x}}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{- 1}{x + h} + \frac{1}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{- x + x + h}{h x \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{h}{h x \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{1}{x \left( x + h \right)}\]
\[ = \frac{1}{x . x}\]
\[ = \frac{1}{x^2}\]
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