= Solution
Expand before taking the fourth <derivative>:
$$
\frac{\sin(x^2)}x=x-\frac{x^5}{6}+\frac{x^9}{120}-\frac{x^{13}}{5040}+O(x^{17}).
$$
The apparent singularity at zero is removable. The convergent power series may be differentiated termwise, giving
$$
y(x)=-20x+\frac{126}{5}x^5-\frac{143}{42}x^9+O(x^{13}).
$$
At $x=0.1$, the successive contributions are $-2$, $0.000252$ and approximately $-3.405\times10^{-9}$. Thus
$$
\boxed{y(0.1)\simeq-1.9997480034.}
$$
Even the first two nonzero terms give $-1.999748$; the next retained correction quantifies that estimate's accuracy.
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