Continuity and ...

If $$(f(x))^{g(y)} = e^{f(x) - g(y)}$$ then $$\dfrac {dy}{dx} =$$.

If $$x + y = \tan^{-1}y$$ and $$y'' =f(y) y'$$ then $$f(y) =$$

If $$y=\sec^{-1}\dfrac{\sqrt{x}+1}{\sqrt{x}-1}+\sin^{-1}\dfrac{\sqrt{x}-1}{\sqrt{x}+1}$$, then $$\dfrac{dy}{dx}=$$

Let $$f$$ be differentiable $$(x\epsilon R)$$, if $$f(2) = -2$$ and $$f'(x) \geq 2$$ for $$x\epsilon [1, 6]$$, then

If $$y = \sin^{-1}\dfrac {1}{2}(\sqrt {1 + x} + \sqrt {1 - x})$$ then $$y' =$$

Let $$g: [1, 6]\rightarrow [0, \infty]$$ be a real valued differentiable function satisfying $$g'(x) = \dfrac {2}{x + g(x)}$$ and $$g(1) = 0$$, the maximum value of $$g$$ cannot exceed.

Let $$f:(-1,1)\rightarrow R$$ be the differentiable function with $$f(0)=-1$$ and $$f'(0)=1$$. 

Determine the value of k for which the following function is continuous at $$x=3$$.

Let $$f(x)=4$$ and $$f'(x)=4$$, then $$\displaystyle \lim _{ x\rightarrow 2 }{ \cfrac { xf(2)-2f(x) }{ x-2 }  } $$

If $$y = Tan^{-1} \left (\dfrac {\log (e/x^{2})}{\log ex^{2}}\right ) + Tan^{-1} \left (\dfrac {3 + 2\log x}{1 - 6\log x}\right )$$ then $$\left (\dfrac {dy}{dx}\right )_{x = 2} + \left (\dfrac {dy}{dx}\right )_{x = 3}$$.