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Question
\[\frac{1}{x^3}\]
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Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{1}{(x + h )^3} - \frac{1}{x^3}}{h}\]
\[ = \lim_{h \to 0} \frac{x^3 - (x + h )^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{x^3 - x^3 - 3 x^2 h - 3x h^2 - h^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{- 3 x^2 h - 3x h^2 - h^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{h\left( - 3 x^2 - 3xh - h^2 \right)}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{\left( - 3 x^2 - 3xh - h^2 \right)}{(x + h )^3 x^3}\]
\[ = \frac{- 3 x^2}{x^6}\]
\[ = \frac{- 3}{x^4}\]
\[ = - 3 x^{- 4} \]
\[\]
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