Differentiation...

Suppose $$A=\displaystyle \frac{dy}{dx}$$ when $$x^2+y^2=4$$ at $$(\sqrt{2},\sqrt{2})$$,$$ B=\displaystyle \frac{dy}{dx}$$ when $$\sin y+ \sin x=\sin x-\sin y$$ at $$(\pi,\pi)$$ and $$C=\displaystyle  \frac{dy}{dx}$$ when $$2e^{xy}+e^x e^y-e^x-e^y=e^{xy+1}$$ at $$(1,1)$$, then $$(A+B+C)$$ has the value equal to 

If $$f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}$$, then the value of $$10 f'(102^+)$$ is

$$\displaystyle y=\left ( 1+\frac{1}{x} \right )^{x}+x^{1+\frac{1}{x}}.$$
Differentiate

The number of distinct rational numbers x such that $$\displaystyle 0 < x < 1$$ and $$\displaystyle x = \frac{p}{q}r$$, where $$\displaystyle p,q \epsilon \left \{ 1,2,3,4,5,6 \right \}$$,is

The number of ways in which three numbers in A.P. can be seleced from the set of first n natural number if n is odd is

If $$\displaystyle y=\sum _{ r=1 }^{ x }{ \tan ^{ -1 }{ \frac { 1 }{ 1+r+{ r }^{ 2 } }  }  } $$ then $$\displaystyle \frac { dy }{ dx } $$ is equal to

If $$\displaystyle \frac { \cos ^{ 4 }{ \theta  }  }{ x } +\frac { \sin ^{ 4 }{ \theta  }  }{ y } =\frac { 1 }{ x+y } $$ then $$\displaystyle \frac { dy }{ dx } =$$

In a test there were n questions. In the test $$\displaystyle 2^{n-i}$$ students gave wrong answers to i questions where $$\displaystyle i=1,2,3...,n$$. If the total number of wrong answers given is 2047 then n is

If the prime sign (') represents differentiation w.r.t. $$x$$ and $$f^{'}=\sin x+\sin 4x.\cos x$$, then $$f^{'}\left ( 2x^{2}+\cfrac{\pi }{2} \right )$$ at $$x=\sqrt{\dfrac{\pi }{2}}$$ is equal to

If $$y=\left | \cos x \right |+\left | \sin x \right |$$ then $$\frac{dy}{dx}$$ at $$x=\frac{2\pi }{3}$$ is