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§2.2 求导法则与导数公式
2.2.1 函数的和、差、积、商的求导法则
定理1 设函数u=u(x)及v=v(x)在点x处可导,C为常数,则
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00077001.jpg?sign=1736288997-i8i8YxaMcVEF7W2C7Zgm8jP8euwaIHQK-0-5f74c9114a15a6804a6a5535e7f65eba)
下面只证明(2),其余留给读者作为练习.
证 由于可导必连续,有
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00077002.jpg?sign=1736288997-OvIBj1BL6vT9l9gRn75EvrxrgcUbxEmt-0-2c39efb45306d8ae543f8f5189592450)
例1 求函数y=tanx的导数.
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00077003.jpg?sign=1736288997-7AAf0O9e6BjDgVIHuxD4dlWAOkwsmc7V-0-ffaba095b69f7daeff0f872cc22ed26f)
即 (tanx)′=sec2x.
类似可得
(cotx)′=-csc2x.
例2 求函数y=secx的导数.
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00077004.jpg?sign=1736288997-Ph7m0dGgXc1OgDLPeS9HWBKJiqsT6Tiz-0-dbdec93a442a4df497b925ecba995861)
即 (secx)′=secxtanx.
类似可得
(cscx)′=-cscxcotx.
例3 设y=3x3+5x2-4x+1,求y′.
解 y′=3(x3)′+5(x2)′-4(x)′+1′=9x2+10x-4.
例4 设,求
解 f′(x)=3x2-3(excosx)′=3x2-3(excosx-exsinx)
=3x2-3ex(cosx-sinx).
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00078001.jpg?sign=1736288997-nuXgOggEr9jBcdOPCpUe08pw1HZBfywX-0-62c75e246d666210a6c5b35b8a2b018f)
例5 设f(x)=x2lnx,求f′(x).
![](https://epubservercos.yuewen.com/578B36/14615859905722306/epubprivate/OEBPS/Images/img00078002.jpg?sign=1736288997-M6vldx4i0KELFNkymLWG7b08qd1ZRuPJ-0-9f1bd4e5b8baa2d358b49f4d0e32023c)