Differentiation of Implicit Functions

Find $\frac{dy}{dx}$ at $x =-1$, when ${\left(\sin y\right)}^{\sin \frac{\pi x}{2}}+\frac{\sqrt{3}}{2}{\sec }^{-1}\left(2x\right)+2^x \tan \left\{{\log }_e\left(x+2\right)\right\} = 0$.

If $y=\sqrt{\cos \left(x\right)+\sqrt{\cos \left(x\right)+\sqrt{\cos \left(x\right)+....\infty }}}$, then prove that $\frac{dy}{dx}=\frac{\sin \left(x\right)}{1-2y}$.

If $x\sin \left(a+y\right)+\sin \left(a\right)\cos \left(a+y\right)=0$. Prove that $\frac{dy}{dx}=\frac{{\displaystyle {\sin }^2}\left(a+y\right)}{\sin \left(a\right)}$

If $x^2+y^2=t-\frac{1}{t}$  and $x^4+y^4=t^2+\frac{1}{t^2}$, then prove that $\frac{dy}{dx}=\frac{1}{x^{3}y}$.

If $\sin \left(y\right)=x\sin \left(a+y\right)$, then prove that $\frac{dy}{dx}=\frac{{\displaystyle {\sin }^2}\left(a+y\right)}{\sin \left(a\right)}$. 

If ${\cos }^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)={\tan }^{-1}\left(a\right)$, $a>0$ then prove that $\frac{dy}{dx}=\frac{y}{x}$.

If ${\log }_e\left(x^2+y^2\right)=2{\tan }^{-1}\left(\frac{y}{x}\right)$, then show that $\frac{dy}{dx}=\frac{x+y}{x-y}$.

If $\sqrt{y+x}+\sqrt{y-x}=c$ (where $c\ne 0$), then $\frac{dy}{dx}$ has the value equal to:

If $2^x+2^y=2^{x+y}$, then $\frac{dy}{dx}$ is equal to:

If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\dots \infty }}}}$, then $\frac{dy}{dx}$ is equal to: