Differentiation...

If $$x\sqrt {1 + y}  + y\sqrt {1 + x}  = 0$$, then $$\dfrac {dy}{dx}$$ is equal to

Seven person $$P_1,P_2......, P_7$$ initially seated at chairs $$C_1,C_2,.....C_7$$ respectively.They all left there chairs simultaneously for hand wash. Now in how many ways they can again take seats such that no one sits on his own seat and $$P_1$$, sits on $$C_2$$ and $$P_2$$ sits on $$C_3$$ ?

If $$m$$ denotes the number of $$5$$ digit numbers if each successive digits are in their descending order of magnitude and $$n$$ is the corresponding figure. When the digits and in their ascending order of magnitude then $$(m-n)$$ has the value

There are $$2$$ identical white balls, $$3$$ identical red balls and $$4$$ green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is

If $$\int_{0}^{x}(t^{2}+2t+2)$$ dt, $$2\leq x\leq 4$$

If $$y^{y^{y^{....{^\infty}}}} = \log_e(x+\log_e(x+....))$$, then $$\dfrac{dy}{dx}$$ at $$(x= e^2-2, y= \sqrt2)$$ is

Let $$y=x^{x^x}$$, then differentiate $$y$$ w.r.t $$x$$.

if $$\int \dfrac {\cos 4x+1}{\cos x-\tan x}dx=A \cos 4x+B;$$ where $$A$$ & $$B$$ are constants, then 

How many $$10-digit$$ numbers can be formed by using the digits $$1$$ and $$2$$?

What is the value of $$^nC_n$$?