If $x^{x}=y^{y}$,then $\frac{d y}{d x}$ is

If $\sqrt{y + x} + \sqrt{y - x} = c$ and $\frac{dy}{dx} = K - \sqrt{\frac{y^2}{x^2} - 1}$, then the value of $K$ is

If $y = x^2 + \frac{1}{x^2 + \frac{1}{x^2 + \frac{1}{x^2 + \dots \infty}}}$,then $\frac{dy}{dx}$ is equal to

Let $f: R \rightarrow R$ be a differentiable function with $f(0)=1$ and satisfying the equation $f(x+y)=f(x) \cdot f^{\prime}(y)+f^{\prime}(x) \cdot f(y)$ for all $x, y \in R$. Then the value of $\log (f(4))$ is

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

If $x^{1/2} y^{1/3} = (x + y)^n$ and $x \frac{dy}{dx} - y = 0$, then $n =$