Finding Maclaurin series grade 4

Question

I've tried finding the Maclaurin series grade 4 of the function:

$cos(x^2)$

I calculated the four derivatives of the function manually and failed somewhere along the way. Is there an easier way to finding the series instead of calculating all the derivatives manually?

EDIT: My derivatives:

$f'(x) = -sin(x^2)*2x$

$f''(x) = -cos(x^2)*4x^2-sin(x^2)$

$f'''(x) = sin(x^2)*8x^3 - 10x*cos(x^2)$

$f''''(x) = cos(x^2)*16x^4 + sin(x^2)*24x^2 - cos(x^2) + 20x^2*sin(x^2)$

Answer 1

You can use the known Maclaurin series for $\cos{x}$ and just substitute $x^2$ instead of $x$.