\displaystyle f(x)=\cos x^2\Big/\thinspace'\\
f'(x)=-2x\sin x^2\Big/\thinspace'\\
f''(x)=-2\sin x^2 -4x^2\cos x^2\Big/\cdot x\\
xf''(x)=-2x\sin x^2-4x^3\cos x^2\\
xf''(x)=f'(x)-4x^3f(x)\Big/\thinspace^{(n-1)}\\
\sum_{i=0}^{n-1}\binom{n-1}i(x)^{(i)}(f'')^{(n-1-i)}(x)=(f')^{(n-1)}(x)-\sum_{i=0}^{n-1}\binom{n-1}i(4x^3)^{(i)}f^{(n-1-i)}(x)\\
\sum_{i=0}^{n-1}\binom{n-1}i(x)^{(i)}f^{(n-i+1)}(x)=f^{(n)}(x)-\sum_{i=0}^{n-1}\binom{n-1}i(4x^3)^{(i)}f^{(n-i-1)}(x)\\
x=0\Rightarrow f^{(n)}(0)=-4(n-1)(n-3)f^{(n-4)}(0)\\
n=100\Rightarrow f^{(100)}(0)=-4\cdot 99\cdot 97\cdot f^{(96)}(0)=-4\cdot 99\cdot 97\cdot (-4)\cdot 95\cdot 93\cdot f^{(92)}(0)=\cdots=(-4)^{25}\cdot 99!!\cdot f^{(0)}(0)=-4^{25}\cdot 99!!