Answer 1

\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right)^3 + 4 \left( x + h \right)^2 + 3\left( x + h \right) + 2 - \left( x^3 + 4 x^2 + 3x + 2 \right)}{h}\]
\[ = \lim_{h \to 0} \frac{x^3 + 3 x^2 h + 3x h^2 + h^3 + 4 x^2 + 4 h^2 + 8xh + 3x + 3h + 2 - x^3 - 4 x^2 - 3x - 2}{h}\]
\[ = \lim_{h \to 0} \frac{3 x^2 h + 3x h^2 + h^3 + 4 h^2 + 8xh + 3h + 2}{h}\]
\[ = \lim_{h \to 0} \frac{h\left( 3 x^2 + 3xh + h^2 + 4h + 8x + 3 \right)}{h}\]
\[ = \lim_{h \to 0} \left( 3 x^2 + 3xh + h^2 + 4h + 8x + 3 \right)\]
\[ = 3 x^2 + 8x + 3\]