设 $\left\{\begin{array}{l}x=\cos \left(t^{2}\right), \\ y=t \cos \left(t^{2}\right)-\int_{1}^{t^{2}} \frac{1}{2 \sqrt{u}} \cos u \mathrm{~d} u,\end{array}\right.$ 求 $\frac{\mathrm{d} y}{\mathrm{~d} x}, \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ 在 $t=\sqrt{\frac{\pi}{2}}$ 的值.