Differentiate the function with respect to $x$: $\cos (\sin x)$

Which of the following statements could be true if $f''(x) = x^{1/3}$?

$I$. $f'(x) = \frac{3}{4}x^{4/3} + 9$	$II$. $f(x) = \frac{9}{28}x^{7/3} - 2$
$III$. $f(x) = \frac{9}{28}x^{7/3} + 6$	$IV$. $f'(x) = \frac{3}{4}x^{4/3} - 4$

$f(x)$ and $g(x)$ are differentiable functions such that $\frac{f(x)}{g(x)} = c$,where $c$ is a non-zero constant. If $\frac{f^{\prime}(x)}{g^{\prime}(x)} = \alpha(x)$ and $\left(\frac{f(x)}{g(x)}\right)^{\prime} = \beta(x)$,then $\frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = $

If $f(x) = \sqrt{x^2 + 1}$, $g(x) = \frac{x + 1}{x^2 + 1}$, and $h(x) = 2x - 3$, then find the value of $f'(h'(g'(x)))$.

Find the derivative of $f$ given by $f(x) = \tan^{-1} x$,assuming it exists.

If $x \neq 0$ and $f(x)$ satisfies $8 f(x) + 6 f(\frac{1}{x}) = x + 5$,then the value of $\frac{d}{dx} (x^2 f(x))$ at $x = 1$ is