Find the coefficient on x^44 for the Taylor series based at 0 where f(x)=ln(2+3x^4)

Please show steps!

In order to do this, we need to get a Taylor series for f(x). Before hand though, notice that if we write f as a composite function, then

f = ln( 2+ 3z), where z = x^4

so, if we obtain a taylor expansion in terms of z, then it will be equivalent to the taylor expansion in powers of x^4. so, instead of the 44th term (33 of which would be 0!), we only need the 11th term of the z-expansion.

Now,
f = ln(2+3z), f(0) = ln(2)

f' = 3/(2+3z), f'(0) = 3/2

f'' = (3)^2 * (-1)/(2+3z)^2, f''(0) = -(3/2)^2

similarly, f'''(0) = (-1)(-2) * (3/2)^3, and

f^(11) (0) = (-1)(-2)(...)(-10) * (3/2)^11 = (-1)^10 * 10! * (3/2)^11

now, we need to divide this by 11! to make it a Taylor coefficient. this gives

(10!/11!) * (3/2)^11 = (3/2)^11/11

since (-1)^10 = 1, and 10!/11! = 1/11.

This is the coefficient of z^11, and thus of (x^4)^11 = x^44.