How do you find the derivative of  arcsin(1/x)?

Feb 3, 2016

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

Explanation:

$\frac{d}{\mathrm{dx}} \arcsin \left(x\right) = \frac{1}{\sqrt{1 - {x}^{2}}}$

For $\arcsin \left(\frac{1}{x}\right)$ use the chain rule. Differentiate inside the bracket then multiply that by the derivative for the function surrounding the bracket so:

$\frac{d}{\mathrm{dx}} \left\{\frac{1}{x}\right\} = - \frac{1}{x} ^ 2$

$\frac{d}{\mathrm{dx}} \arcsin \left(\frac{1}{x}\right) = - \frac{1}{x} ^ 2 \frac{1}{\sqrt{1 - {\left(\frac{1}{x}\right)}^{2}}}$

And now simplify the denominator:

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