Question

What is the derivative of this function `y=sin^-1(1/x)`?

Answer 1

`(-1)/(xsqrt(x^2-1)`

Explanation:

`color(orange)"Reminder" d/dx(sin^-1x)=1/(sqrt(1-x^2)`

here, however, x = `1/x`

Differentiate using the`color(blue)" chain rule combined with power rule"`

`color(orange)" Chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|)))........ (A)`
`"---------------------------------------------------------------"`

`f(g(x))=sin^-1(1/x)rArrf'(g(x))=1/(sqrt(1-(1/x)^2)`

and `g(x)=1/x=x^-1rArrg'(x)=-x^-2=-1/x^2`
`"-----------------------------------------------------------------"`
Substitute these values into (A)

`=1/(sqrt(1-1/x^2))xx-1/x^2`

`=(-1)/(x^2sqrt(1/x^2(x^2-1))`

`=(-1)/(x^2xx1/xsqrt(x^2-1)`

`rArrd/dx(sin^-1(1/x))=(-1)/(xsqrt(x^2-1)`