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

Let $f$ and $g$ be two differentiable functions on $R$ such that $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then for all $x$,which of the following is true?

If $2 f(x)-3 f\left(\frac{1}{x}\right)=x+1$,then $f^{\prime}(\sqrt{3})$ is equal to

Let $h(x)$ be differentiable for all $x$ and let $f(x) = (kx + e^x) h(x)$ where $k$ is some constant. If $h(0) = 5$,$h'(0) = -2$,and $f'(0) = 18$,then the value of $k$ is equal to

$\frac{d}{dx} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) = $ . . . . . .

If $f(x) = e^x g(x)$,$g(0) = 2$,and $g'(0) = 1$,then $f'(0)$ is