If $x^2+y^2=4$, then $y \frac{dy}{dx}+x=$

If $y=\sqrt{\cosh x+\sqrt{\cosh x+\dots}}$,then $\frac{d y}{d x}=$

If $x = y\sqrt{1 - y^2}$,then $\frac{dy}{dx} = $

Find $\frac{dy}{dx}$ for the function $y^{x} = x^{y}$.

Let $f$ be a differentiable function such that $(x - y) f(x + y) - (x + y) f(x - y) = 4xy(x^2 - y^2)$ and $f(1) = 2$. Then the area enclosed by $\frac{|f(x) - x|^{1/3}}{17} + \frac{|f(y) - y|^{1/3}}{2} \le \frac{1}{4}$ is

Find $\frac{dy}{dx}$ for the equation $2x + 3y = \sin y$.