Find the Maclaurin series for f(x) = cos(x^2) and use it to determine f^ (8)(0).

Question

Find the Maclaurin series for

f(x) = cos(x²)

and use it to determine

f ⁽⁸⁾(0).

Answer 1

cosx = 1 + x²/2! + x⁴/4! + x⁶/6! + x⁸/8! + ...

So, cos(x²) = 1 + x⁴/2! + x⁸/4! + x¹²/6! + ...

If f(x) = cos(x²), then the 8th derivative of f(x) evaluated at x = 0 is:

(1/4!)(8!) + 0 + 0 + 0 + ... = (8)(7)(6)(5) = 1680