곱의 미분

[출제의도] 다항함수의 미분 이해하기 $\displaystyle \lim\limits _{x \rightarrow 1} {\frac{f \left( x \right) - 3}{x - 1}} = 2$이고 $\displaystyle \lim\limits _{x \rightarrow 1} {\left( x - 1 \right)} = 0$이므로 $\displaystyle \lim\limits _{x \rightarrow 1} {\left\{ f \left( x \right) - 3 \right\}} = 0$에서 $f \left( 1 \right) = 3$ $\displaystyle \lim\limits _{x \rightarrow 1} {\frac{f \left( x \right) - 3}{x - 1}} = \lim\limits _{x \rightarrow 1} {\frac{f \left( x \right) - f \left( 1 \right)}{x - 1}} = f' \left( 1 \right)$이므로 $f' \left( 1 \right) = 2$ $g' ( x ) = 3 x ^{2} f \left( x \right) + x ^{3} f' \left( x \right)$이므로 $g' \left( 1 \right) = 3 f \left( 1 \right) + f' \left( 1 \right) = 3 \times 3 + 2 = 11$