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

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SUMMARY

The discussion focuses on simplifying the Taylor series calculation for the function f(x) = cos(3x²) at x = 0, specifically computing P5(x). The Maclaurin series formula is applied, and the user initially struggles with higher-order derivatives. However, by substituting u = 3x², they successfully derive the series up to the fourth degree, resulting in 1 - 9x⁴/2, which is confirmed as correct. The user ultimately resolves their query by recognizing the need to stop at the fourth degree.

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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
 
Last edited:
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Yes, you did. And a good job too.
 

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