Question

x的y次幂=y的x次幂,求y的导数.

Answer 1

取对数后求隐函数导数,然后提取y'为主项.

x^y=y^x

ylnx=xlny

(ylnx)'=(xlny)'

lnx*y'+y*(lnx)'=lny*x'+x*(lny)'

y'lnx+y/x=lny+xy'/y

y'lnx-xy'/y=lny-y/x

y'(lnx-x/y)=lny-y/x

y'=(lny-y/x)/(lnx-x/y)再进一步化简

y'=[(xlny-y)/x]/[(ylnx-x)/y]

y'=(xylny-y²)/(xylnx-x²)