Continuity and ...

Given that $$f(x)$$ is a differentiable function of $$x$$ and that $$f\left( x \right).f\left( y \right) = f\left( x \right) + f\left( y \right) + f\left( {xy} \right) - 2$$ and that $$f\left( 2 \right) = 5.$$ Then $$f'\left( 3 \right)$$ is equal to

If $$y^{y^{y^{....{^\infty}}}} = \log_e(x+\log_e(x+....))$$, then $$\dfrac{dy}{dx}$$ at $$(x= e^2-2, y= \sqrt2)$$ is

Let $$y=x^{x^x}$$, then differentiate $$y$$ w.r.t $$x$$.

If $$x = 2\left( {\theta  + \sin \theta } \right)and\,y = 2\left( {1 - \cos \theta } \right),then\;value\;of\frac{{dy}}{{dx}}\;is\;$$

If $$f\left( x \right) = x + \left| x \right| + {\kern 1pt} \,\cos \left( {\left[ {{\pi ^2}} \right]x} \right)\,\,and\,\,g\left( x \right) = \,\sin \,x\,where\,\left[ . \right]$$ denotes the greatest integer function) then :-
$$(1)$$ $$f\left( x \right) + g\left( x \right)$$ is discontinuous
$$(2)$$$$f\left( x \right) + g\left( x \right)$$ is differentiable everywhere  
$$(3)$$ $$f\left( x \right) \times g\left( x \right)$$  is differentiable everywhere
 $$(4)$$ $$f\left( x \right) \times g\left( x \right)$$ is countimuos but not diffrentiable  at $$x=0$$ 

If $$f\left( x \right) = \left\{ \begin{array}{l}\frac{{1 - \left| x \right|}}{{1 + x}},{\rm{ }}x \ne  - 1\\1,{\rm{          }}x =  - 1{\rm{     }}\end{array} \right.$$   then $$f\left( {\left[ {2x} \right]} \right),$$ where $$\left[ {} \right]$$ represents the greatest integer function , is 

if $$y^{2}=ax+bx+c$$, then $$ y^{3} \dfrac {d^{2}y}{dx^{2}}$$ is

 If $$f:\left[ {0,1} \right] \to \left[ {0,1} \right]$$ be definded by f(x) =\begin{cases} x,\quad \quad \quad \quad \quad \quad if\quad x\quad is\quad rational\quad  \\ 1-x,\quad \quad \quad \quad if\quad x\quad is\quad irrational\quad \quad \quad \quad \quad  \end{cases} then $$\left( {f \circ f} \right)x$$ ______________.

If $$\left( \frac { 1-x }{ 1+x }  \right) =x$$ and $$g\left( x \right) =\int { f\left( x \right) } dx$$ then 

If $$f(x)=\begin{cases} \dfrac { 1-\sqrt { 2 } \sin { x }  }{ \pi -4x } ,\quad \quad ifx\neq \dfrac { \pi  }{ 4 }  \\ a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\quad \quad ifx=\dfrac { \pi  }{ 4 }  \end{cases}$$ is continous at $$x=\dfrac {\pi}{4}$$ then $$a=$$