Compute the derivative of $g(x) = \cot x$.

If $y = f\left( \frac{2x - 1}{x^2 + 1} \right)$ and $f'(x) = \sin(x^2),$ then $\frac{dy}{dx} = $

If $y = \tan^{-1} \left[ \frac{4 \sin 2x}{\cos 2x - 6 \sin^2 x} \right]$,then $\frac{dy}{dx}$ at $x = 0$ is

Let $f: R \rightarrow R$ satisfy the equation $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and $f(x) \neq 0$ for any $x \in R$. If the function $f$ is differentiable at $x=0$ and $f'(0)=3$,then $\lim_{h \rightarrow 0} \frac{1}{h}(f(h)-1)$ is equal to ....... .

If $f(x)=e^{x} g(x)$,$g(0)=4$,and $g^{\prime}(0)=2$,then $f^{\prime}(0)=$

Which one of the following is false?