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