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1 X 3 - Mathematics

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\[\frac{1}{x^3}\]

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Solution

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

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 1.03 | Page 25

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