微分教學兩題

1.f(x)=[x(x－1)(x－2)]/(x－5)，f(0)一次微分
Sol
依題意f(0)一次微分為0，改為求f’(0)
f’(0)=lim(x->0)_[f(x)－f(0)]/x
=lim(x->0)_[x(x－1)(x－2)/(x－5)－0]/x
=lim(x->0)_(x－1)(x－2)/(x－5)
=2/5

2.y=(x－47)/(x+1)，y(1)一次微分
Sol
依題意y(1)一次微分為0，改為求y’(1)
y’(1)=lim(x->1)_[y(x)－y(1)]/(x－1)
=lim(x->1)_[(x－47)/(x+1)－(1－47)/2]/(x－1)
=lim(x->1)_[(x－47)/(x+1)+23]/(x－1)
=lim(x->1)_[(x－47)+23(x+1)]/[(x+1)(x－1)]
=lim(x->1)_(24x－24)/[(x+1)(x－1)]
=lim(x->1)_24/(x+1)
=12

連乘或是連除的函數 我習慣取對數

1.f(x)=x(x-1)(x-2)/(x-5)
ln[f(x)]=ln(x)+ln(x-1)+ln(x-2)-ln(x-5)~再微分
f '(x)/f(x)=1/x+1/(x-1)+1/(x-2)-1/(x-5)
f '(x)=f(x)*[1/x+1/(x-1)+1/(x-2)-1/(x-5)]
=(x-1)(x-2)/(x-5)+x(x-2)/(x-5)+x(x-1)/(x-5)-x(x-1)(x-2)/(x-5)^2
f '(0)=-2/5
~這個方法有個好處是後面含有x的項都是零

2.y=(x-47)/(x+1)~同樣方法
ln(y)=ln(x-47)-ln(x+1)
y'/y=1/(x-47)-1/(x+1)
y'=y*[1/(x-47)-1/(x+1)]=1/(x+1)-(x-47)/(x+1)^2
y'|x=1=1/2-(-46)/4
=24/2=12

此外這個方法再遇見更多項時很好用

例如f(x)=x(x-1)(x-2)....(x-9)(x-10) 求 f '(1)~這要用傳統微分就很頭痛
但是用對數就方便多了

f '(x)=f(x)*[ln(x)+ln(x-1)+...ln(x-10)]
=(x-1)(x-2)..(x-10)+x(x-2)....(x-10)+........+x(x-1)..(x-9)~除了劃線項外都是零
f '(1)=1(-1)(-2)....(-9)=-9!

1. f '(x) = { (x-5)[(x-1)(x-2)+x(x-2)+x(x-1)] - x(x-1)(x-2) } / (x-5)^2
=[(x-5)(x^2-3x+2+x^2-2x+x^2-x) - (x^3-3x^3+2x)] / (x-5)^2
=[(x-5)(3x^2-6x+2)-(x^3-3x^3+2x)] / (x-5)^2
f '(0) = [(-5)(2)-0] / (-5)^2 = -2/5

2. y ' = (x+1)*1-1*(x-47) / (x+1)^2
=48/(x+1)^2
y '(1)=48/2^2=48/4=12

2012-03-06 10:50:30 補充：
[x(x-1)(x-2)]微分=x ' (x-1)(x-2)+x(x-1) ' (x-2)+ x(x-1)(x-2) '
f(x)/g(x)微分=g(x)f '(x) - g '(x)f(x) / [g(x)]^2