Ответ:

[tex]y = \arcsin{ \frac{x - 1}{x} }[/tex]

[tex]y' = \frac{d}{dx} ( \arcsin{ \frac{x - 1}{x} })[/tex]

[tex]y' = \frac{d}{dg} ( \arcsin{g}) \times \frac{d}{dx} ( \frac{x - 1}{x} )[/tex]

Найдём производную:

[tex] \frac{d}{dg} ( \arcsin{g}) \\ \frac{1}{ \sqrt{1 - {g}^{2} } } [/tex]

Вычислим производную частного:

[tex] \frac{d}{dx} ( \frac{x - 1}{x} ) \\ \frac{ \frac{d}{dx} (x - 1) \times x - (x - 1) \times \frac{d}{dx}(x) }{ {x}^{2} } \\ \frac{1x - (x - 1) \times 1}{ {x}^{2} } \\ \frac{x - (x - 1)}{ {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - {g}^{2} } } \times \frac{x - (x - 1)}{ {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - {( \frac{x - 1}{x}) }^{2} } } \times \frac{x - (x - 1)}{ {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - {( \frac{x - 1}{x} )}^{2} } } \times \frac{x - x + 1}{ {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - {( \frac{x - 1}{x} )}^{2} } } \times \frac{1}{ {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - {( \frac{x - 1}{x}) }^{2} } {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{1 - \frac{ {(x - 1)}^{2} }{ {x}^{2} } } {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \sqrt{ \frac{ {x}^{2} - {(x - 1)}^{2} }{ {x}^{2} } } {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \frac{ \sqrt{ {x}^{2} - {(x - 1)}^{2} } }{ |x| } \times {x}^{2} } [/tex]

[tex]y' = \frac{1}{ \frac{ \sqrt{ {x}^{2} - {(x - 1)}^{2} } {x}^{2} }{ |x| } } [/tex]

[tex]y' = \frac{ |x| }{ \sqrt{ {x}^{2} - {(x - 1)}^{2} } {x}^{2} } [/tex]

[tex]y' = \frac{ |x| }{ \sqrt{ {x}^{2} - ( {x}^{2} - 2x + 1) } {x}^{2} } [/tex]

[tex]y' = \frac{ |x| }{ \sqrt{ {x}^{2} - {x}^{2} + 2x - 1 {x}^{2} } } [/tex]

[tex]y' = \frac{ |x| }{ \sqrt{2x - 1} {x}^{2} } [/tex]