If $$y$$ is a function of $$x$$ and log $$(x+y)-2xy=0$$, then the value of $$y'(0)$$ is equal to

Let $$f:\left( {0,\infty } \right) \to R$$ and $$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.} $$ If $$F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$$, then $$f(4)$$ equals

If $${x^2} + {y^2} = 1$$ then

If $$y = {\left( {\sin x} \right)^{\tan x}},$$ then $${{dy} \over {dx}}$$ is equal to