If $y = (\tan^{-1} x)^2$,then $(x^2 + 1)^2 \frac{d^2 y}{dx^2} + 2x(x^2 + 1) \frac{dy}{dx} = $

$A$ polynomial $P(x)$ with real coefficients has the property that $P^{\prime \prime}(x) \neq 0$ for all $x$. Suppose $P(0) = 1$ and $P^{\prime}(0) = -1$. What can you say about $P(1)$?

If $\frac{d^2x}{dy^2} \left( \frac{dy}{dx} \right)^3 + \frac{d^2y}{dx^2} = K$,then the value of $K$ is equal to

Let $f(x)$ and $g(x)$ be two functions having finite non-zero $3^{rd}$ order derivatives $f'''(x)$ and $g'''(x)$ for all $x \in R$. If $f(x)g(x) = 1$ for all $x \in R$,then $\frac{f'''}{f'} - \frac{g'''}{g'}$ is equal to

If $y = \frac{(\sin^{-1} x)^2}{2}$, then $(1-x^2) y_2 - x y_1 = $

If $y=f(x)$ is a twice differentiable function such that at a point $P$,$\frac{dy}{dx}=4$ and $\frac{d^2y}{dx^2}=-3$,then $\left(\frac{d^2x}{dy^2}\right)_P$ is equal to: