Continuity and ...

Let $$f$$ differentiable at $$x=0$$ and $$f'\left( 0 \right) = 1$$. Then $$\mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( h \right) - f\left( { - 2h} \right)}}{h} = $$=

Let $$f(x)$$ be differentiable function such that $$f\left(\dfrac{x+y}{1-xy}\right)=f(x)+f(y) \forall x$$ and $$y$$. If $$\underset { x\rightarrow 0 }{ lt } \dfrac { f\left( x \right)  }{ x } =\dfrac { 1 }{ 3 }$$ then $$f(1)$$ equals 

If $$f (x)$$ is differentiable everywhere, then:

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

If $$f(x)=x^{n}\sin \dfrac {1}{x},f(0)=0$$

If $$x=\dfrac{e^t+e^{-t}}{2}, y=\dfrac{e^t-e^{-t}}{2}$$, then $$\dfrac{dx}{dy}=$$

If $$y=a^{{a}^{{x}}}$$, then $$\dfrac {dy}{dx}=$$

If f (x + y) = 2 f(x) f(y) all x, y  $$\in$$ R where f' (0) = 3 and f (4) =2, then f'(4) is equal to 

If $$(2+\sin x)\dfrac{dy}{dx}+(y+1)\cos x=0$$ and $$y(0)=1$$, then $$y\left(\dfrac{\pi}{2}\right)$$ is equal to?

If $$(\cos x)^{y}=(\sin y)^{x}$$, then $$\dfrac{dy}{dx}=$$