Continuity and ...

Let $$f(x + y) = f(x) f(y)$$ for all $$x$$ and $$y$$. If $$f(0) = 1, f(3) = 3$$ and $$f'(0) = 11$$, then $$f'(3)$$ is equal to

Let $$f(x+y) = f(x) f(y) $$ and $$f(x) = 1+\sin(3x)g(x)$$, where $$g$$ is differentiable. The $$f'(x)$$ is equal to

If $$y=\cos^{-1}x^{3}$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\cos^{-1}(4x^{3}-3x)$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\tan^{-1}\left(\dfrac{a\cos x-b\sin x}{b\cos x+a\sin x}\right)$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\tan^{-1}\left(\dfrac{\sqrt{a+\sqrt{x}}}{1-\sqrt{ax}}\right)$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\tan^{-1}(\sec x+6\tan x)$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\sin^{-1}(3x-4x^3)$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\tan^{-1}\left\{\dfrac{\cos x+\sin x}{\cos x-\sin x}\right\}$$ then $$\dfrac{dy}{dx}=?$$

Which of the following statement is always true ([ .] represents the greatest integer function)