第5章 5.9

(1)

$y = x^{\frac{2}{3}}$

$y' = \frac{2}{3}x^{-\frac{1}{3}} = \frac{2}{3\sqrt[3]{x}}$

(2)

$y = (3x+1)^{\frac{1}{2}}$

$y' = \frac{3}{2}(3x+1)^{-\frac{1}{2}}$

(3)

$y = (x+3)^{\frac{1}{3}}$

$y' = \frac{1}{3}(x+3)^{-\frac{2}{3}}$

(4)

$y = (x^2+2)^{\frac{1}{2}}$

$y' = \frac{1}{2}(x^2+2)^{-\frac{1}{2}} \cdot 2x = x(x^2+2)^{-\frac{1}{2}}$

(5)

$y' = \frac{\dfrac{1}{2\sqrt{x}}(x+1) - \sqrt{x}}{(x+1)^2} = \frac{\dfrac{x+1-2x}{2\sqrt{x}}}{(x+1)^2} = \frac{-x+1}{2\sqrt{x}(x+1)^2}$

(6)

$y' = \frac{0 - \dfrac{1}{\sqrt{2x+1}}}{\sqrt{2x+1}^{\,2}} = -\frac{1}{(2x+1)^{\frac{3}{2}}}$