Question

由方程y^x=x^y所确定的隐函数y=y(x)的导数dy/dx

Answer 1

解法一：对数求导法
y^x = x^y
x lny = y lnx，两边求导
lny + x/y•dy/dx = lnx•dy/dx + y/x
(x/y - lnx)•dy/dx = y/x - lny
(x - ylnx)/y•dy/dx = (y - xlny)/x
dy/dx = [y(y - xlny)]/[x(x - ylnx)]

解法二：链式法则
y^x = x^y
d(y^x)/dy•dy/dx + d(y^x)/dx•dx/dx = d(x^y)/dx•dx/dx + d(x^y)/dy•dy/dx
x•y^(x - 1)•dy/dx + (y^x)ln(y) = y•x^(y - 1) + (x^y)ln(x)•dy/dx
[(x/y)(y^x) - (x^y)ln(x)]•dy/dx = (y/x)(x^y) - (y^x)ln(y)
dy/dx = [(y/x)(x^y) - (y^x)ln(y)]/[(x/y)(y^x) - (x^y)ln(x)]
= y•[x(y^x)ln(y) - y(x^y)] / { x•[y(x^y)ln(x) - x(y^x)] }

Answer 2

取对数
xlny=ylnx
求导
lny+x*1/y*y'=y'*lnx+y*1/x
(x/y-lnx)y'=y/x-lny
所以dy/dx=(y/x-lny)/(x/y-lnx)