Differentiating Functions with Rational Exponents

1. \(\dfrac{d}{dx}\sqrt{x}=\dfrac{d}{dx}x^{1/2}=\dfrac{1}{2} x^{( 1/2) -1}=\dfrac{1}{2}x^{-1/2}=\dfrac{1}{2x^{1/2}}=\dfrac{1}{2 \sqrt{x}}=\dfrac{\sqrt{x}}{2x} \)
2. \(\dfrac{d}{du}\sqrt[3]{u}=\dfrac{d}{du}u^{1/3}=\dfrac{1}{3} u^{-2/3}=\dfrac{1}{3\sqrt[3]{u^{2}}}=\dfrac{\sqrt[3]{u}}{3u}\)
3. \(\dfrac{d}{dx}x^{5/2}=\dfrac{5}{2}x^{3/2}\)
4. \(\dfrac{d}{ds}s^{-3/2}=-\dfrac{3}{2}s^{-5/2}=-\dfrac{3}{2s^{5/2}}\)