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Question
x2 + x + 3
Solution
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right)^2 + x + h + 3 - \left( x^2 + x + 3 \right)}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 + h^2 + 2xh + x + h + 3 - x^2 - x - 3}{h}\]
\[ = \lim_{h \to 0} \frac{h^2 + 2xh + h}{h}\]
\[ = \lim_{h \to 0} \frac{h(h + 2x + 1)}{h}\]
\[ = \lim_{h \to 0} h + 2x + 1\]
\[ = 0 + 2x + 1\]
\[ = 2x + 1\]
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