第6章 6.1

(1)

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

(2)

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

(3)

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

(4)

$y'=2x\cdot\log x+x^2\cdot\frac{1}{x}=2x\log x+x$

(5)

$y'=\frac{0-\left(1+\dfrac{1}{x}\right)}{(x+\log x)^2}=-\frac{\dfrac{x+1}{x}}{(x+\log x)^2}=\frac{x+1}{x(x+\log x)^2}$

(6)

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