[Maximum mark: 22]

The function $f$ is defined by �(�)=�6�cos⁡�f(x)=e6xcosx, for $x\in \mathbb{R}$.

1. 1. Find the derivative of $f(x)$.
   2. Find the second derivative of $f(x)$.
   3. Hence find the Maclaurin series for $f(x)$ up to and including the �2x2 term.[6]
2. Show that the coefficient of �3x3 in the Maclaurin series for $f(x)$ is $33$.                                                  [7]
3. Using the Maclaurin series for $\sin x$ and $\ln (1+x)$, find the Maclaurin series for $\ln (1+\sin (3x))$ up to and including the 3x3 term.                                                 [6]
4. Hence, or otherwise, find lim⁡�→0�(�)−1ln⁡(1+sin⁡(3�))x→0lim​ln(1+sin(3x))f(x)−1​.                                                 [3]