Continuity and ...

If $$x = a \left (\cos t + \log \tan \dfrac {t}{2}\right ), y = a\sin t$$, then $$\dfrac {dy}{dx} =$$

If $$f(x)=\cos ^{ -1 }{ \left\{ \cfrac { 1-{ \left( \log _{ e }{ x }  \right)  }^{ 2 } }{ 1+{ \left( \log _{ e }{ x }  \right)  }^{ 2 } }  \right\}  } $$, then $$f'(e)$$

If $$y=\tan ^{ -1 }{ \left( \cfrac { a\cos { x } -b\sin { x }  }{ b\cos { x } +a\sin { x }  }  \right)  } $$, then $$\cfrac { dy }{ dx } $$ is equal to

If $$x=\cos { \theta  } ,y=\sin { 5\theta  } $$, then
$$(1-{ x }^{ 2 })\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -x\cfrac { dy }{ dx } =$$

If $$y=x\log { \left( \cfrac { x }{ a+bx }  \right)  } $$, then $${ x }^{ 3 }\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } $$ is equal to

If $$ u = 2 (t - \sin t ) $$ and $$ v = 2 (1- \cos t), $$ then $$ \dfrac {dv}{du} $$ at $$ t = \dfrac {2 \pi}{3} $$ is equal to :

Let $$f(x)=2\tan^{-1}x+\sin^{-1}\left(\displaystyle\frac{2x}{1+x^2}\right)$$. Then

If $${ y }^{ x }-{ x }^{ y }=1$$, then the value of $$\cfrac { dy }{ dx } $$ at $$x=1$$ is

The value of a for which the function $$f(x)=\left\{\begin{matrix} \tan^{-1} a-3x^2, & 0<x<1 \\ -6x, & x\geq 1\end{matrix}\right.$$ has a maximum at $$x=1$$, is?

If $$ y^x = 2^x , $$ then $$ \dfrac {dy}{dx} $$ is equal to :