1.f(x)=sin^2(3x - -2) 求f '(x) Ans:3sin(6x-4)
Ans:
基本微分公式: (u^n)'=n*u^(n-1)*u'
f'(x)=2*sin(3x-2)*[sin(3x-2)]'
=2*sin(3x-2)*cos(3x-2)*(3x-2)'
=2*sin(3x-2)*cos(3x-2)*3
=6*sin(3x-2)*cos(3x-2)
=3*sin(2(3x-2))......sin2Q=2cosQsinQ
=3*sin(6x-4)
=版主答案
2.f(x)=acot[(1+x)/(1-x)] 求f '(x) Ans:-1/1+x^2
Ans:
基本微分公式: [acot(u)]'=-u'/(1+u^2)
u=(1+x)/(1-x)
u'=[(1-x)*1-(1+x)*(-1)]/(1-x)^2
=(1-x+1+x)/(1-x)^2
=2/(1-x)^2
f'(x)=-u'/(1+u^2)
=[-2/(1-x)^2]*1/[1+(1+x)^2/(1-x)^2]
=-2/{(1-x)^2*[1+(1+x)^2/(1-x)^2]}
=-2/[(1-x)^2+(1+x)^2]
=-2/(1-2x+x^2+1+2x+x^2)
=-2/(2+2x^2)
=-1/(1+x^2)
=版主答案
3.f(x)=x√(a^2-x^2)+a^2*asin(x/a) 求f '(x) Ans:2√(a^2-x^2)
Ans:
基本微分公式: [asin(u)]'=u'/√(1-u^2)
u=x/a => u'=1/a
f'(x)=√(a^2-x^2)-x*2x/2√(a^2-x^2)+a^2*(1/a)/√(1-u^2)
=√(a^2-x^2)+x^2/√(a^2-x^2)+a/√(1-x^2/a^2)
=√(a^2-x^2)+x^2/√(a^2-x^2)+a^2/√(a^2-x^2)
=√(a^2-x^2)+(x^2+a^2)/√(a^2-x^2)
=√(a^2-x^2)+√(a^2-x^2)
=2√(a^2-x^2)
=版主答案
4.f(x)=1/ab tan^-1〔b/a tan x 〕求f '(x) Ans:1/a^2 cos^2 x+b^2 sin^2 x
Ans:
基本微分公式: [atan(u)]'=u'/(1+u^2)
f'(x)=(1/ab)*[(b/a)tan(x)]'/[1+(b/a)^2*tan^2(x)]
=(a/b)*(b/a)*sec^2(x)/[a^2+b^2*tan^2(x)]
=sec^2(x)/[a^2+b^2*tan^2(x)]
=1/{cos^2(x)[a^2+b^2*tan^2(x)]}
=1/[a^2*cos^2(x)+b^2*sin^2(x)]
=版主答案
5.f(x)=[In(x+3)]^2 求f '(x) Ans: 2 In(x+3)/x+3
Ans:
基本微分公式: [ln(u)]'=u'/u
f'(x)=2*ln(x+3)[ln(x+3)]'
=2*ln(x+3)(x+3)'/(x+3)
=2*ln(x+3)*1/(x+3)
=2*ln(x+3)/(x+3)
=版主答案
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