Question

x/(1－x)＋(2x-x²）/(1-x)²的导数？

Answer 1

首先，将式子展开得到：
x/(1-x) + (2x-x^2)/(1-x)^2 = x/(1-x) + 2x/(1-x)^2 - x^2/(1-x)^2
然后，对每一项分别求导：
d/dx [x/(1-x)] = (1-x)1/(1-x)^2 - x(-1)/(1-x)^2 = 1/(1-x)^2
d/dx [2x/(1-x)^2] = 2*(1-x)^2*(-1)/(1-x)^3 + 2x2(1-x)*1/(1-x)^4 = -2/(1-x)^2 + 4x/(1-x)^3
d/dx [-x^2/(1-x)^2] = -2x*(-1)/(1-x)^3 + x^22(1-x)*1/(1-x)^4 = 2x/(1-x)^3 - 2x^2/(1-x)^4
将三个部分加起来得到：
d/dx [x/(1-x) + (2x-x^2)/(1-x)^2] = 1/(1-x)^2 - 2/(1-x)^2 + 4x/(1-x)^3 + 2x/(1-x)^3 - 2x^2/(1-x)^4
化简后得到：
d/dx [x/(1-x) + (2x-x^2)/(1-x)^2] = (x^3 - 3x^2 + 2x)/(1-x)^4
因此，原式的导数为 (x^3 - 3x^2 + 2x)/(1-x)^4。