If $x^3 + y^3 = 6$ and $\frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2} = \frac{m}{(xy)^n}$, where $m, n \in R$, then find the value of $\frac{m}{n}$.

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is

For $n \in \mathbb{N}$,find the $n$-th derivative of $\log x$,i.e.,$\frac{d^{n}}{d x^{n}}(\log x) = $

If $f(x) = a\sin (\log x)$,then ${x^2}f''(x) + xf'(x) = . . . $

If $y_k$ is the $k$-th derivative of $y$ with respect to $x$,and $y = \cos(\sin x)$,then $y_1 \sin x + y_2 \cos x$ is equal to

If $x = \int\limits_0^y {\frac{{dt}}{{\sqrt {1 + {t^2}} }}} $,then $\frac{{{d^2}y}}{{d{x^2}}}$ is equal to