How Can You Simplify the Taylor Series Calculation for cos(3x^2)?

[Kaura]

Homework Statement

Determine the Taylor series for the function below at x = 0 by computing P₅(x)
f(x) = cos(3x²)

Homework Equations

Maclaurin Series for degree 5

f(0) + f¹(0)x + f²(0)x²/2! + f³(0)x³/3! + f⁴(0)x⁴/4! + f⁵(0)x⁵/5!

The Attempt at a Solution

I know how to do this but attempting to solve the 3rd derivative of cos(3x²) and onward is simply infeasible due to it requiring multiplication rule and stuff
I remember my professor mentioning some sort of short cut to certain series
Is there a short cut or heuristic to solve this or do I simply have to solve the higher order derivatives?

Update
I tried solving the series as cos(u) where u = 3x² and got
1 - 9x⁴/2 + 27x⁸/8
which matches the result from a Taylor Series calculator online
I feel like I am making a basic mistake right now please enlighten me

Update
Genius me did not realize that I needed to stop at the 4th degree even after doing to replacement
1 - 9x⁴/2
was accepted as the correct answer
I guess I ended up answering my own question

 

[LCKurtz]

Yes, you did. And a good job too.