Differentiation...

If $$y=logx^3+3 sin^{-1}x+kx^2$$, then find $$\displaystyle \frac {dy}{dx}$$

If $$\displaystyle y=\frac { x }{ a+\displaystyle\frac { x }{ b+\displaystyle\frac { x }{ a+\displaystyle\frac { x }{ b+.....\infty  }  }  }  } $$, then $$\cfrac{dy}{dx} =$$

Given : $$f(x)=4x^3-6x^2\cos2a+3x \sin 2a.\sin 6a+\sqrt{\ln (2a-a^2)}$$ then 

If $$y = sec^{-1}\left(\displaystyle\frac{\sqrt x + 1}{\sqrt x - 1}\right) + \sin^{-1}\left(\displaystyle\frac{\sqrt x - 1}{\sqrt x + 1}\right)$$, then $$\displaystyle\frac{dy}{dx}$$ equals

The solution set of $${f}'(x)>{g}'(x)$$ where $$f(x)=\displaystyle \frac{1}{2}(5^{2x+1})$$ & $$g(x)= 5^x+4x(\ln 5)$$ is 

$$f:R\rightarrow R$$ and $$\displaystyle f(x)=\frac {x(x^4+1)(x+1)+x^4+2}{x^2+x+1}$$, then $$f(x)$$ is

Let $$\displaystyle f\left( \frac { { x }_{ 1 }+{ x }_{ 2 }+...+{ x }_{ n } }{ n }  \right) =\frac { f\left( { x }_{ 1 } \right) +f\left( { x }_{ 2 } \right) +...+f\left( { x }_{ n } \right)  }{ n } $$ where all $${ x }_{ i }\in R$$ are independent to each other and $$n\in N$$. if $$f(x)$$ is differentiable and $$f'\left( 0 \right) =a,f\left( 0 \right) =b$$ and $$f'\left( x \right) $$ is equal to

Suppose the function $$f(x)-f(2x)$$ has the derivative $$5$$ at $$x=1$$ and derivative $$7$$ at $$x=2$$.The derivative  of the function $$f(x)-f(4x)$$ at $$x=1$$, has the value equal to 

$$y=\sqrt{\sin x+\sqrt{\sin x +\sqrt{\sin x+-\infty }}}$$ then $$\displaystyle \frac{dy}{dx}$$ equals:$$(\sin x> 0)$$

If P(x) is a polynomial such that $$P\left ( x^{2}+1 \right )=\left \{ P\left ( x^{2} \right ) \right \}^{2}+1$$ and $$P(0)=0$$ then $$P^{'}(0)$$ is equal to