Question

Differentiate
w.r.t. x.
((x^2+1)/x)^1/2​

Answer 1

Question:

Differentiate w.r.t.x

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

Answer:

Let,

$y = ({ \frac{ {x}^{2} + 1}{x} })^{ \frac{1}{2} } \\ \\ diff \: w.r.t.x \\ \\ \frac{dy}{dx} = \frac{1}{2}( { \frac{ {x}^{2} + 1 }{x} })^{ \frac{ - 1}{2} } \frac{d}{dx} ( \frac{ {x}^{2} + 1 }{x} ) \\ \\ \frac{dy}{dx} = \frac{1}{2} ({ \frac{ {x}^{2} + 1 }{x} })^{ \frac{ - 1}{2} } \times \frac{x. \frac{d}{dx}( {x}^{2} + 1) -( {x}^{2} + 1). \frac{d}{dx} (x)}{ {x}^{2} } \\ \\ \frac{dy}{dx} = \frac{1}{2}( { \frac{ {x}^{2} + 1 }{x} })^{ \frac{ - 1}{2} } \times \frac{x.(2x )- ( {x}^{2} + 1)(1) }{ {x}^{2} } \\ \\ \frac{dy}{dx} = \frac{1}{2} ({ \frac{ {x}^{2} + 1 }{x} })^{ \frac{ - 1}{2} } \times \frac{2 {x}^{2} - {x}^{2} - 1}{ {x}^{2} } \\ \\ \frac{dy}{dx} = \frac{1}{2} ({ \frac{ {x}^{2} + 1 }{x} })^{ \frac{ - 1}{2} } \times \frac{ {x}^{2} - 1 }{ {x}^{2} } \\ \\ \frac{dy}{dx} = \frac{1}{2} (\frac{x}{ {x}^{2} + 1}) ^{ \frac{1}{2} } \times \frac{ {x}^{2} - 1}{ {x}^{2} }$

Formula used:

$\star \: \frac{d}{dx} ( {x}^{n} ) = n {x}^{n - 1} \\ \\ \star \: \frac{d}{dx} ( \frac{u}{v} ) = \frac{v \frac{d}{dx} (u) - u \frac{d}{dx} (v)}{ {v}^{2} }$