If $f(x) = \sin x \cdot \sin 2x \cdot \sin 3x$ and $f''(x) = a(\sin bx) + c(\sin dx) + e(\sin kx)$,then the value of $(a+c+e) - (b+d+k)$ equals

If $y=\log \left(x-\sqrt{x^2-1}\right)$,then $\left(x^2-1\right) y^{\prime \prime}+x y^{\prime}$ is equal to:

If $x+1=e^{-y}$,then $\frac{d^2 y}{d x^2} = $ . . . . . .

If $y = x^2 e^{mx}$,where $m$ is a constant,then $\frac{d^3y}{dx^3} = $

If $m \sin ^{-1} x = \log _{e} y$,then $(1 - x^{2}) y'' - x y'$ is equal to

Let $f$ and $g$ be twice differentiable functions such that $f(x) \cdot g(x) = 1$ for all $x \in R$ and $f'$ and $g'$ are never zero. Then $\frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)}$ equals: