If $f(2)=4$ and $f^{\prime}(2)=1$,then $\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}$ is equal to

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

Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a, b \in R$,$g^{\prime}(a) = 5$ and $g(a) = b$,then $f^{\prime}(b)$ is equal to

Let $f: R \rightarrow R$ be defined by $f(x) = \log \left[e^x \left(\frac{x-2}{x+2}\right)^{3/4}\right]$. Find the value of $f'(0)$.

Cosine of the angle of intersection of curves $y = 3^{x - 1} \log x$ and $y = x^x - 1$ is

The derivative of $F[f\{ \phi (x)\} ]$ with respect to $x$ is: