If $f(x)$ is differentiable on $R$,$f(x) f^{\prime}(-x) - f(-x) f^{\prime}(x) = 0$,$f(0) = 3$,and $f(3) = 9$,then $(1 + f(-3))^3 + 1 = $

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

If $x^y \cdot y^x = 16$,then $\frac{dy}{dx}$ at $(2, 2)$ is

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

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

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int_{0}^{x} f(t) \,d t$ for $0 \leq x \leq 1$ and $f(0)=0$,then $\lim_{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) \,d t$ is: