Calculus 10th Edition

Rewrite: $x^{-\frac{1}{2}}$; Differentiate: $(-\frac{1}{2})(x^{-\frac{1}{2}-1})$; Simplify: $-\frac{1}{2x\sqrt x}$.
To simplify remember the index rule ($\frac{x^a}{x^b}=x^{a-b}$); hence the function becomes $x^{\frac{1}{2}-1}$ which is $x^{-\frac{1}{2}}$. To differentiate, just use the power rule to get $(-\frac{1}{2})(x^{-\frac{1}{2}-1})$. To simplify, use the index rule ($x^{-n}=\frac{1}{x^n}$). Then, since the power is an improper fraction, we rewrite $\sqrt{x^3}$ as $\sqrt{x^2x}$ which becomes $x\sqrt{x}$; hence, the final simplified derivative is $-\frac{1}{2x\sqrt x}$.