If $y = \sin x + e^x,$ then $\frac{d^2x}{dy^2} = $

$A$ non-zero polynomial with real coefficients has the property that $f''(x) f'(x) = f(x)$. Then the value of $f'''(x)$ is

Let $f(x) = \tan^{-1} x$. Then $f'(x) + f''(x) = 0$ when $x$ is equal to:

Let $f : S \rightarrow S$ where $S =(0, \infty)$ be a twice differentiable function such that $f(x+1) = xf(x)$. If $g : S \rightarrow R$ is defined as $g(x) = \log_{e} f(x)$,then the value of $|g''(5) - g''(1)|$ is equal to:

If $y(\theta) = \frac{2 \cos \theta + \cos 2 \theta}{\cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2}$,then at $\theta = \frac{\pi}{2}$,$y'' + y' + y$ is equal to:

If $p(x)$ is a polynomial of degree $3$ which satisfies $p^{\prime \prime}(1)=0$ and $p^{\prime \prime \prime}(1)=6$,then $p^{\prime \prime}(0)$ is equal to