If $a y^4 = (x + b)^5$,then $\frac{y \cdot (\frac{d^2 y}{d x^2})}{(\frac{d y}{d x})^2} = $

If $y = \tan^{-1} \sqrt{\frac{1 - \cos x}{1 + \cos x}}$,then the values of $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ respectively are

If $\log _e y=3 \sin ^{-1} x$,then $(1-x^2) y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to :

If $f(x)$ and $g(x)$ are twice differentiable functions satisfying $f''(x) = g''(x)$, $f'(1) = 2g'(1) = 4$ and $f(2) = 3g(2) = 9$, then the value of $f(x) - g(x)$ at $x = 4$ is:

If $f(x)$ is a function such that $f^{\prime \prime}(x)+f(x)=0$ and $g(x)=[f(x)]^{2}+[f^{\prime}(x)]^{2}$ and $g(3)=8$,then $g(8)$ is equal to

If the function $y = \sin^{-1} x$,then $\left(1-x^2\right) \frac{d^2 y}{d x^2}$ is equal to