Finding Taylor Series - different Method

[JaeKyung]

Homework Statement

Hello, I'm in the middle of solving for the Taylor series of the function:

f(x)=sin(2x)ln(1-x)

up to n = 4.

The Attempt at a Solution

So far, I've been strictly taking its derivatives until I reach the fourth.
It's becoming a very long process considering it's just one of the many homework problems, and I can't help but think if there's a more elegant way of doing this.

Thank you in advance.

 

[HallsofIvy]

Since you are only required to do this for n up to 4, I would suggest that you write out the Taylor's polynomials for sin(2x) and ln(1- x) separately (you should be familiar enough with those that you don't need to actually do the derivatives) and then multiply.