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(一) 反正(余)割的导数问题反正(余)割的导数有如下两种处理方法: (Ⅰ) 规定反正割函数的主值区间为[0,π/2)∪(π/2,π]这样在整个主值区间内都有arccos1/x=arcsecx。此时不难证明:(arcsecx)′1/|x|(x~2-1)~(1/2) [1]文就是这样规定反正割的主值区间的,但是由于忽视了算术根,它所得到的结论:(arcsecx)′=1/x(x~2-1)~(1/2)是错误的。(见[1]272页)。 (Ⅱ) 规定反正割函数的主值区间为:
(a) The derivatives of the (remainder) cuts have the following two treatments for the derivatives of the (remainder) cuts: (I) The main value interval specified for the arc secant function is [0, π/2) ∪ (π/2, π] This has arccos1/x=arcsecx in the entire main value interval, and it is not difficult to prove that: (arcsecx)’1/|x|(x~2-1)~(1/2)[1] This stipulates the secant-cut main value interval, but because of neglecting the arithmetic root, it concludes that (arcsecx)’=1/x(x~2-1)~(1/2) is wrong. [1] page 272. (II) The main value interval specified for the arc secant function is: