Question

一道微积分题目.y=ln(1/3 (1+e^(-2x) ))求证 dy/dx=2/3 (e^(-y)-3)

Answer 1

y=ln(1/3 (1+e^(-2x) ))

3e^y = 1+e^(-2x)

3e^y·e^2x = e^2x +1

两边求导

(3e^ydy)·e^2x + (2e^2xdx)·3e^y = 2e^2xdx

3e^2x·e^y·dy = (2e^2x - 6e^2x·e^y)dx

3e^ydy = (2 - 6e^y)dx

dy/dx = 2/3(e^(-y) - 3)