If $f(x) = x \tan^{-1} x$,then $\lim_{x \rightarrow 1} \frac{f(x) - f(1)}{x - 1}$ is equal to

Let for a differentiable function $f:(0, \infty) \rightarrow \mathbb{R}$,$f(x)-f(y) \geq \log_e\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty)$. Then $\sum_{n=1}^{20} f^{\prime}\left(\frac{1}{n^2}\right)$ is equal to

Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+6, x \in R$,then $f(2)$ equals

$y=\log \left\{\left(\frac{1+x}{1-x}\right)^{1 / 4}\right\}-\frac{1}{2} \tan ^{-1}(x)$,then $\frac{d y}{d x}$ is equal to

If $y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \infty$,then $\frac{dy}{dx} = $

Let $f$ and $g$ be differentiable functions satisfying $g'(a) = 2$,$g(a) = b$,and $f \circ g = I$ (identity function). Then $f'(b)$ is equal to