Differentiation...

If   $$\displaystyle y^{x}=x^{\sin y} $$, find $$\cfrac{dy}{dx}$$.

The numbers 1, 2, ..., 100 are arranged in the squares of an table in the following way: the numbers 1, ... , 10 are in the bottom row in increasing order,  numbers 11, ... ,20 are in the next row in increasing order, and so on. One can choose any number and two of its neighbors in two opposite directions (horizontal, vertical, or diagonal). Then either the number is increased by 2 and its neighbors are decreased by 1, or the number is decreased by 2 and its neighbors are increased by 1. After several such operations the table again contains all the numbers 1, 2, ... , 100. Prove that they are in the original order.

$$\displaystyle y=(\cot x)^{\sin x}+(\tan  x)^{\cos x}$$.Find dy/dx 

$$\displaystyle \frac{d}{dx}(\log_{e}\left ( \frac{1+x}{1-x} \right )^{1/4}-\frac{1}{2}\tan^{-1}x.)$$

$$\displaystyle \frac{d}{dx}\tan ^{-1}\left(\frac{a \cos x-b\sin x}{b\cos x+a\sin x}\right)$$

$$\displaystyle \frac{d}{dx}(\tan ^{-1}\frac{\sin x+\cos x}{\cos x-\sin x})$$

If  $$5f(x)+3f\left ( \displaystyle \frac{1}{x} \right )=x+2$$ and $$y=xf(x)$$ then $$\left (\displaystyle  \frac{dy}{dx} \right )_{x=1}$$ is equal to ?

If $$\displaystyle  f\left( x \right) =\sqrt { 1+\sqrt { x }  } , x > 0,$$ then $$\displaystyle f\left ( x \right )\cdot f'\left ( x \right )$$ is equal to

Rajdhani Express going from Bombay to Delhi stops at five inter-mediate stations, $$10$$ passengers enter the train during the journey with $$10$$ different ticket of two classes. The number of different sets of tickets they may have is

$$\displaystyle \dfrac{d}{dx}\tan ^{-1}\left(\dfrac{\cos x}{1+\sin x}\right)$$