Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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(X + 2)3 - Mathematics

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(x + 2)3

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Solution

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

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 1.11 | Page 25

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