3.2.2 基本初等函數(shù)的導(dǎo)數(shù)公式及導(dǎo)數(shù)的運(yùn)算法則(二)
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學(xué) 習(xí) 目 標(biāo)
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核 心 素 養(yǎng)
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1.理解函數(shù)的和、差、積、商的求導(dǎo)法則.
2.能夠綜合運(yùn)用導(dǎo)數(shù)公式和導(dǎo)數(shù)運(yùn)算法則求函數(shù)的導(dǎo)數(shù).(重點(diǎn)、難點(diǎn))
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借助導(dǎo)數(shù)公式及運(yùn)算法則求函數(shù)的導(dǎo)數(shù),培養(yǎng)數(shù)學(xué)運(yùn)算素養(yǎng).
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導(dǎo)數(shù)的運(yùn)算法則
(1)設(shè)兩個(gè)函數(shù)f(x),g(x)可導(dǎo),則
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和的導(dǎo)數(shù)
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[f(x)+g(x)]′=f′(x)+g′(x)
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差的導(dǎo)數(shù)
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[f(x)-g(x)]′=f′(x)-g′(x)
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積的導(dǎo)數(shù)
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[f(x)·g(x)]′=f′(x)g(x)+f(x)g′(x)
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商的導(dǎo)數(shù)
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=(g(x)≠0)
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(2)常數(shù)與函數(shù)的積的導(dǎo)數(shù)
[cf(x)]′=cf′(x)(c為常數(shù))
思考:根據(jù)商的導(dǎo)數(shù)的運(yùn)算法則,試求函數(shù)y=的導(dǎo)數(shù).
[提示] y′===-.

1.函數(shù)y=x·ln x的導(dǎo)數(shù)是( )
A.x B.
C.ln x+1 D.ln x+x
C [y′=(x)′×ln x+x×(ln x)′=ln x+1.]
2.函數(shù)y=x4+sin x的導(dǎo)數(shù)為( )
A.y′=4x3 B.y′=cos x
C.y′=4x3+sin x D.y′=4x3+cos x
D [y′=(x4)′+(sin x)′=4x3+cos x.]
3.函數(shù)y=的導(dǎo)數(shù)為__________.
y′=- [y′==-.]

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利用導(dǎo)數(shù)的運(yùn)算法則求導(dǎo)數(shù)
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【例1】 求下列函數(shù)的導(dǎo)數(shù):
(1)y=+sin cos ;
(2)y=x+2;
(3)y=cos xln x;
(4)y=.
[解] (1)y′=
=(x-2)′+
=-2x-3+cos x
=-+cos x.
(2)y′=
=(x3)′--(6x)′+(2)′
=3x2-3x-6.
(3)y′=(cos xln x)′
=(cos x)′ln x+cos x(ln x)′
=-sin xln x+.
(4)y′====.