# Find the Maclaurin Series for f(x) = cos(x^{2}) and use it to Determine f^{(8)}(0).

Maclaurin series is nothing but Taylor series about the point x = 0.

## Answer: cos (x^{2}) = 1 + x^{4}/2! + x^{8}/4! + x^{12}/6! + x^{16}/8! + ..., f^{(8)}(0) = 1680.

Let us write the Maclaurin series for f(x) = cos(x^{2}) and use it to determine f^{(8)}(0).

**Explanation:**

Maclaurin series of the function f is given by

f(x) = f(0) + f'(0)x + f"(0) x^{2} / 2! + f'"(0) x^{3} / 3! + ... + f^{(n)}(0) x^{n} / n! + ... --- (1)

Using the definition of Maclaurin series, we can write cos x as

cos x = 1 + x^{2}/2! + x^{4}/4! + x^{6}/6! + x^{8}/8! + ...

Replace x by x^{2}, we get

cos (x^{2}) = 1 + x^{4}/2! + x^{8}/4! + x^{12}/6! + x^{16}/8! + ... --- (2)

f^{(8)}(0) is the 8^{th} derivative of f(x) evaluated at x = 0.

Comparing the coefficients of x^{8} from equations (1) and (2), we get

f^{(8)}(0) / 8! = 1/4!

f^{(8)}(0) = 8! / 4! = 1680