第1章 1.6

(1)

$y = \sin^{-1} 5x$

$y' = \frac{(5x)'}{\sqrt{1-25x^2}} = \frac{5}{\sqrt{1-25x^2}}$

(2)

$y = \tan^{-1} 3x$

$y' = \frac{(3x)'}{9x^2+1} = \frac{3}{9x^2+1}$

(3)

$y = \sin^{-1} \dfrac{x}{3}$

$y' = \frac{\left(\dfrac{x}{3}\right)'}{\sqrt{1-\dfrac{x^2}{9}}} = \frac{\dfrac{1}{3}}{\sqrt{\dfrac{9-x^2}{9}}} = \frac{1}{\sqrt{9-x^2}}$

(4)

$y = \tan^{-1} \sqrt{x}$

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