Continuity and ...

If $$y=1-\cos\theta,x=1-\sin\theta$$, then $$\dfrac{dy}{dx}$$ at $$\theta=\dfrac{\pi}{4}$$ is 

If $$x=\sqrt{2^{cosec^{-1}t}}$$ and $$y=\sqrt{2^{sec^{-1}t}}(|t|\geq 1)$$ then $$\dfrac{dy}{dx}$$ is equal to:

For  $$x > 1 ,$$  if  $$( 2 x ) ^ { 2 y } = 4 e ^ { 2 x - 2 y } ,$$  then   $$\left( 1 + \log _ { \mathrm { e } } 2 \mathrm { x } \right) ^ { 2 } \dfrac { \mathrm { dy } } { \mathrm { dx } }$$  is equal to :

$$\dfrac{d}{dx}\left[\tan h^{-1}\left(\dfrac{2x}{1+x^2}\right)\right]=?$$

A differential function satisfies equation $$f(x)=\int_{0}^{x}(f(t)\cos\ t-\cos(t-x))dt$$ then

If  $$f ( x )$$  and  $$g ( x )$$  are differentiable functions in  $$[ 0,1 ]$$  such that  $$f ( 0 ) = 2 , f ( 1 ) = 6 , g ( 0 ) = 0 , g ( 1 ) =2$$   then there exists  $$0 < c < 1$$  such that

If $$ y = \sin ^ { - 1 } x$$ then $$ \frac { d y } { d x } $$ is equal to 

If $$f(x)=0$$ for $$x < 0$$ and $$f(x)$$ is differential at $$x=0$$ then for $$x\ge 0, f(x)$$ may be  

If $$y={ \tan }^{ -1 }{ ax}$$ then, which of the following is true

If $$f(x)$$ is a differentiable function $$\forall \ x\ \epsilon \ R$$ so that, $$f(2)=4,\ f'(x)\ge 5\ \forall \ x\ \epsilon \ [2,6]$$, then, $$f(6)$$ is :