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

Jul 13, 2016

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

#### Explanation:

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

here, however, x = $\frac{1}{x}$

Differentiate using the$\textcolor{b l u e}{\text{ chain rule combined with power rule}}$

$\textcolor{\mathmr{and} a n \ge}{\text{ Chain rule}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left(f \left(g \left(x\right)\right)\right) = f ' \left(g \left(x\right)\right) g ' \left(x\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(A\right)$
$\text{---------------------------------------------------------------}$

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

and $g \left(x\right) = \frac{1}{x} = {x}^{-} 1 \Rightarrow g ' \left(x\right) = - {x}^{-} 2 = - \frac{1}{x} ^ 2$
$\text{-----------------------------------------------------------------}$
Substitute these values into (A)

$= \frac{1}{\sqrt{1 - \frac{1}{x} ^ 2}} \times - \frac{1}{x} ^ 2$

=(-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)