곱으로 정의된 함수의 미분계수

조건 (가)에서 곡선 $y = f \left( x \right)$ 위의 점 $\left( 1 , 1 \right)$에서의 접선의 기울기는 $12$이므로 $f \left( 1 \right) = 1$, $f' \left( 1 \right) = 12$ $\cdots$$\cdots$ ㉠ 조건 (나)에서 $g \left( x \right)$$\displaystyle = f \left( x \right) \times \sum\limits _{n = 1} ^{4} \frac{x ^{n}}{n}$$\displaystyle = f \left( x \right) \left( x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \frac{x ^{4}}{4} \right)$ 이때 $\displaystyle g' \left( x \right) = \left( 1 + x + x ^{2} + x ^{3} \right) f \left( x \right) + \left( x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \frac{x ^{4}}{4} \right) f' \left( x \right)$ $\therefore$ $g' \left( 1 \right)$$\displaystyle = \left( 1 + 1 + 1 + 1 \right) f \left( 1 \right) + \left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right) f' \left( 1 \right)$ $\displaystyle = 4 f \left( 1 \right) + \frac{25}{12} f' \left( 1 \right)$ $\displaystyle = 4 \times 1 + \frac{25}{12} \times 12$ ($\because$㉠) $= 4 + 25 = 29$