第6章 6.9

(1)

$y'=\dfrac{3x^{2}-3}{x^{3}-3x+1}$

(2)

$y'=\dfrac{1+\dfrac{2x}{2\sqrt{x^{2}+A}}}{x+\sqrt{x^{2}+A}}=\dfrac{\dfrac{\sqrt{x^{2}+A}+x}{\sqrt{x^{2}+A}}}{x+\sqrt{x^{2}+A}}=\dfrac{1}{\sqrt{x^{2}+A}}$

(3)

$y'=(\log x-1)+x\cdot\dfrac{1}{x}=\log x$

(4)

$y=\dfrac{1}{2a}\{\log(x-a)-\log(x+a)\}$

$y'=\dfrac{1}{2a}\left(\dfrac{1}{x-a}-\dfrac{1}{x+a}\right)=\dfrac{1}{2a}\dfrac{(x+a)-(x-a)}{(x-a)(x+a)}=\dfrac{1}{2a}\dfrac{2a}{x^{2}-a^{2}}=\dfrac{1}{x^{2}-a^{2}}$

(5)

$y'=\dfrac{(e^{x}-1)-xe^{x}}{(e^{x}-1)^{2}}=\dfrac{(1-x)e^{x}-1}{(e^{x}-1)^{2}}$

(6)

$y'=(\sin x)'e^{\sin x}=\cos x\cdot e^{\sin x}$