9. If \( y_{1}=\cos x_{1}, y_{2}=\sin x_{1} \cos x_{2}, y_{3}=\sin x_{1} \sin x_{2} \cos x_{3}, \ldots \) and \( y_{n}=\sin x_{1} \sin x_{2} \ldots \sin x_{n-1} \cos x_{n} \), then find \( \frac{\partial\left(y_{1}, y_{2}, \ldots, y_{n}\right)}{\partial\left(x_{1}, x_{2}, \ldots, x_{n}\right)} \).
10. If \( y_{1}=1-x_{1}, y_{2}=x_{1}\left(1-x_{2}\right), y_{3}=x_{1} x_{2}\left(1-x_{3}\right), \ldots \), and \( y_{n} \ldots x_{1} x_{2}=x_{n-1}\left(1-x_{n}\right) \), then find \( \frac{\partial\left(y_{1}, y_{2}, \ldots, y_{n}\right)}{\partial\left(x_{1}, x_{2}, \ldots, x_{n}\right)} \)
11. If \( y_{1}=r \sin \theta_{1} \sin \theta_{2}, y_{2}=r \sin \theta_{1} \cos \theta_{2}, y_{3}=r \cos \theta_{1} \sin \theta_{3} \) and \( y_{4}=r \cos \theta_{1} \cos \theta_{3} \), prove that
\[
\frac{\partial\left(y_{1}, y_{2}, y_{3}, y_{4}\right)}{\partial\left(r, \theta_{1}, \theta_{2}, \theta_{3}\right)}=r^{3} \sin \theta_{1} \cos \theta_{1} \text {. }
\]
