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.
\( 3 x-\frac{9}{2} x^{3}+\frac{81}{40} x^{5}+\cdots \)
\( 3-\frac{3}{2} x^{2}+\frac{5}{8} x^{4}+\cdots \)
\( 1-\frac{27}{2} x^{2}+\frac{135}{8} x^{4}+\cdots \)
\( 3-\frac{27}{2} x^{2}+\frac{81}{8} x^{4}+\cdots \)

Answer

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Answer

\( g^{\prime}(x) = 3 - \frac{9}{2} x^{2} + \frac{81}{40} x^{4} + \cdots \)

Steps

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 ：\( g^{\prime}(x) = 3 - \frac{9}{2} x^{2} + \frac{81}{40} x^{4} + \cdots \)