高数。求下列函数的一阶导数 简单题

Question

如题所述

Answer 1

Answer 2

(1)
y=[x/(1+x)]^x
lny = xln[x/(1+x)]
(1/y)dy/dx =ln[x/(1+x)] + x.[ (1+x)/x ] . [1/(1+x)^2]
=ln[x/(1+x)] + 1/(1+x)
dy/dx =[ln[x/(1+x)] + 1/(1+x)] . [x/(1+x)]^x

(2)
y=[x/(1+x^2)]^x
lny =xln[x/(1+x^2)]
(1/y)dy/dx =ln[x/(1+x^2)] + x.[(1+x^2)/x] . { [(1+x^2)- 2x^2]/(1+x^2)^2 }
=ln[x/(1+x^2)] + (1-x^2)/(1+x^2)
dy/dx ={ ln[x/(1+x^2)] + (1-x^2)/(1+x^2) } . [x/(1+x^2)]^x

Answer 3

两边同时用ln
直接求导就行了

Answer 4

两边ln