How do you find the derivative of (1)/((1-x^2)^(1/2))?

Derivative of $\textcolor{b l u e}{\frac{d}{\mathrm{dx}} \left(\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)\right) = \frac{x}{1 - {x}^{2}} ^ \left(\frac{3}{2}\right)}$

Explanation:

Start with the given $\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)$

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

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

$\frac{d}{\mathrm{dx}} \left(\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)\right) = \left(- \frac{1}{2}\right) {\left(1 - {x}^{2}\right)}^{- \frac{3}{2}} \cdot \left(0 - 2 x\right)$

$\frac{d}{\mathrm{dx}} \left(\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)\right) = \left(- \frac{1}{2}\right) {\left(1 - {x}^{2}\right)}^{- \frac{3}{2}} \cdot \left(- 2 x\right)$

$\frac{d}{\mathrm{dx}} \left(\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)\right) = \frac{x}{1 - {x}^{2}} ^ \left(\frac{3}{2}\right)$

God bless....I hope the explanation is useful.