If $8 f(x)+6 f\left(\frac{1}{x}\right)=x+5$ and $y=x^2 f(x)$,then $\frac{d y}{d x}$ at $x=-1$ equals

If the slope of the tangent to the curve $xy + ax - by = 0$ at the point $(1, 1)$ is $2$,then what are the values of $a$ and $b$ respectively?

If ${x^y} \cdot {y^x} = 1$,then ${{dy} \over {dx}} = $

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 $f(x)$ 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 = $

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: