The derivative of $\tan x - x$ with respect to $x$ is

The differential coefficient of $\frac{{\tan^{-1}x}}{{1 + \tan^{-1}x}}$ with respect to $\tan^{-1}x$ is

Differentiate the following with respect to $x$: $\cos (\log x + e^x)$,where $x > 0$.

If $y=\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\log \sqrt{1-x^2}$,then $\frac{d y}{d x}=$

Differentiate the function with respect to $x$: $\cos (a \cos x + b \sin x)$,where $a$ and $b$ are constants.

Let $f(x) = \mathop {\text{Lim}}\limits_{h \to 0} \frac{1}{h} \int\limits_x^{x + h} \frac{dt}{t + \sqrt{1 + t^2}}$,then $\mathop {\text{Lim}}\limits_{x \to -\infty} x \cdot f(x)$ is