Mathematics - Differentiation Question with Solution | TestHub

At $x=-1,\sin \left(\frac{\pi x}{2}\right)=-1$ and the relation gives $(\sin y{)}^{-1}=-\frac{\sqrt{3}}{2}{\sec }^{-1}(-2)-2^{-1}\tan (1)<0$, so $\sin y<0$. In any neighbourhood of $x=-1$, the exponent $\sin \left(\frac{\pi x}{2}\right)$ takes irrational values, and with $\sin y<0$ the power $(\sin y{)}^{\sin \left(\frac{\pi x}{2}\right)}$ is not real-defined; so $y$ is not defined in any full neighbourhood of $x=-1$ and $\frac{dy}{dx}$ at $x=-1$ does not exist. Statement 2 is true and is exactly the reason statement 1 holds.