Question

$if \lim_{n \to \infty} a_n y = (x+\frac{1}{x} )(\sqrt{x} +\frac{1}{x} ) find \frac{dy}{dx} +?$

Answer 1

Answer:

Step-by-step explanation:

$y=(x+\frac{1}{x} )(x^{1/2}+x^{-1/2}  )$

$\frac{dy}{dx} =\frac{d}{dx} (x+\frac{1}{x})(x^{1/2} +x^{-1/2}  )$

$= (x+\frac{1}{x} )\frac{d}{dx} (x^{1/2} +x^{-1/2} )+(x^{1/2} +x^{-1/)2})\frac{d}{dx}  (x+\frac{1}{x}$

$(x+\frac{1}{x} )(\frac{1}{2} x^{1/2-1} -\frac{1}{2}x^{-1/2-1}  )$ +$x^{1/2} +x^{-1/2} )(\frac{dx}{dx} +\frac{d}{dx} x^{-1}$

$(x+\frac{1}{x} ).\frac{1}{2} x^{-1/2} -\frac{1}{2} x^{-3/2} +(x^{1/2} +x^{-1/2} )$ $(1-1 x^{-1-1} )$

$(x+\frac{1}{2} )(\frac{1}{2}  \frac{1}{x^{1/2} } -\frac{1}{2} \frac{1}{x^{3/2} } )+x^{1/2} +\frac{1}{x^{1/2} } )(1-x^{-2} )$

$(x+\frac{1}{x} )(\frac{1}{2}  \frac{1}{x^{1/2} } -\frac{1}{x^{3/2} } +x^{1/2} \frac{1}{x^{1/2} } )(1-\frac{1}{x^{2} } )$

$(x+\frac{1}{x} )(\frac{1}{2} (\frac{1 }{\sqrt{x} } -\frac{1}{x^{3/2} } )+\sqrt{x} +\frac{1}{\sqrt{x} } )(1-\frac{1}{x^{2} } )$

$\frac{1}{2} (x+\frac{1}{x} )(\frac{1}{\sqrt{x} } -\frac{1}{x^{3/2} } )+(\sqrt{x} +\frac{1}{\sqrt{x} } )(1-\frac{1}{x^{2} } )$

Proved............√√√√√√√√√√√