Let $y$ be an implicit function of $x$ defined by ${x^{2x}} - 2{x^x}\cot y - 1 = 0$. Then $y'(1)$ equals

For $x > 1$,if $(2x)^{2y} = 4e^{2x-2y}$,then $(1 + \log_e 2x)^2 \frac{dy}{dx}$ is equal to

If $x = e^{\tan^{-1}\left(\frac{y-x^2}{x^2}\right)}$,then $\frac{dy}{dx}$ at $x = 1$ is

If $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4$,then $\frac{d y}{d x}=$

If $xy \neq 0, x+y \neq 0$ and $x^m y^n=(x+y)^{m+n}$,where $m, n \notin N$,then $\frac{dy}{dx}$ is equal to

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a twice differentiable function. If for some $a \neq 0$,$\int_0^1 f(\lambda x) d\lambda = a f(x)$,$f(1) = 1$ and $f(16) = \frac{1}{8}$,then $16 - f^{\prime}\left(\frac{1}{16}\right)$ is equal to . . . . . .