What is the slope of the tangent line drawn to the hyperbola $xy = a$ $(a \ne 0)$ at the point $(a, 1)$?

Find the derivative of $e^{x}+e^{y}=e^{x+y}$.

If $\tan y = \frac{x \sin \alpha}{1-x \cos \alpha}$ and $\frac{dy}{dx} = \frac{m}{x^2+2nx+1}$,then $m^2+n^2$ is

If $y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots}}}$,then $\frac{dy}{dx}=$

$f(x)$ and $g(x)$ are two differentiable functions on $[0, 2]$ such that $f''(x) - g''(x) = 0$,$f'(1) = 2$,$g'(1) = 4$,$f(2) = 3$,and $g(2) = 9$. Then $f(x) - g(x)$ at $x = 3/2$ is:

If $2x^2 - 3xy + 4y^2 + 2x - 3y + 4 = 0$, then $\left(\frac{dy}{dx}\right)_{(3,2)} = $