Continuity and ...

If $$y \tan^{-1} \left ( \sqrt{\frac{a - x}{a + x}} \right )$$, where -a < x < a, then $$\frac{dy}{dx} =$$.....

If $$y = \tan^{-1} \left ( \frac{x}{1 + \sqrt{1 - x^2}} \right ) + \sin \left [ 2 \tan^{-1} \left ( \sqrt{\frac{1 - x}{1 + x}} \right )  \right ]$$ then $$\frac{dy}{dx} = $$...........

If $$y = \sin(2 \sin^{-1} x)$$, then dx = .......

If function $$f(x)=\dfrac{x^2-9}{x-3}$$ is continuous at $$x=3$$, then value of $$(3)$$ will be:

If $$f(x)=\begin{cases} \begin{matrix} \dfrac{\log (1+mx)- \log (1-nx)}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} k; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=0$$ then the value of $$k$$ will be:

If function $$f(x)=\begin{cases} \begin{matrix} \dfrac{\sin 3x}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} m; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=2$$ then value of $$m$$ will be:

Let $$x={f}''(t) cost +{f}'(t) sint$$  and $$y={-f}''(t) sint+{f}'(t) cost.$$ Then $$\displaystyle \int \left [ \left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2 \right ]^{\frac{1}{2}} dt$$ equals
(Note : $$f(x), f'(x), f''(x) , f'''(x) >0$$ )

The set of all points where the function $$f(\displaystyle \mathrm{x})=\frac{x}{1+|x|}$$ is differentiable is 

If $$y=x^{\displaystyle x^{\displaystyle x^{\displaystyle \dots^{\displaystyle\infty}}}}$$, find $$\displaystyle\frac{dy}{dx}$$.

Derivative of $$({\log{x}})^{\displaystyle\cos{x}}$$ with respect to $$x$$ is