For the function $y = \tan^{-1} x$,the value of $(1 + x^2) y_2$ is . . . . . .

$A$ 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 $y=f(x)$ 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:

If $f(x) = |x|^3$,show that $f''(x)$ exists for all real $x$,and find it.

Let $p(x)$ be a polynomial such that $p(x) - p'(x) = x^n$,where $n$ is a positive integer. Then,$p(0)$ equals

If $x = \sin \theta$ and $y = \cos(p \theta)$,then $(1 - x^2) y_2 =$