The derivative of ln(x + sin x) with respect to (x + cos x) is

If $$y = {\cot ^{ - 1}}\left[ {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right]$$, where $$0 < x < {\pi \over 2}$$, then $${{dy} \over {dx}}$$ is equal to

The function $$f(x) = {{{x^2}} \over {{e^x}}}$$ is monotonically increasing, if

If $${x^a}{y^b} = {(x - y)^{a + b}}$$, then the value of $${{dy} \over {dx}} - {y \over x}$$ is equal to