Problem
Question 10 ( 4 points) Let \( g(x)=x \cos (3 x) \). Find the Maclaurin series of its derivative, \( g^{\prime}(x) \) None of the other answers are correct. \( 3 x-\frac{9}{2} x^{3}+\frac{81}{40} x^{5}+\cdots \) \( 3-\frac{3}{2} x^{2}+\frac{5}{8} x^{4}+\cdots \) \( 1-\frac{27}{2} x^{2}+\frac{135}{8} x^{4}+\cdots \) \( 3-\frac{27}{2} x^{2}+\frac{81}{8} x^{4}+\cdots \)
Solution
Step 1 ：\( g(x) = x \cos(3x) \)
Step 2 ：\( g^{\prime}(x) = \cos(3x) - 3x \sin(3x) \)
Step 3 ：\( g^{\prime}(0) = 3, g^{\prime\prime}(0) = -\frac{9}{2}, g^{\prime\prime\prime}(0) = \frac{81}{40} \)
Step 4 ：\( g^{\prime}(x) = 3 - \frac{9}{2} x^{2} + \frac{81}{40} x^{4} + \cdots \)
