sinxy的导数怎么求设y是由方程sin(xy)-1/y-x=1所确定的函数，则y'|x=0,（）

Question

sinxy的导数怎么求
设 y是由方程sin(xy)-1/y-x=1 所确定的函数，则y'|x=0,（）

Answer 1

sin(xy)-1/y-x=1
[dsin(xy)/d(xy)*d(xy)/dx]-(-y'/y²)-1=0
cos(xy)*(y+xy')+y'/y²-1=0
ycos(xy)+xy'cos(xy)+y'/y²-1=0
y³cos(xy)+xy²y'cos(xy)+y'-y²=0
y'[xy²cos(xy)+1)=y²-y³cos(xy)
y'=[y²-y³cos(xy)]/[xy²cos(xy)+1]
当x=0,
sin(0*y)-1/y-0=1
-1/y=1
∴y=-1
∴y'|(x=0,y=-1)
=[(-1)²-(-1)³cos(0*(-1))/[0*(-1)²cos(0*(-1))+1]
=(1+1)*1/(0+1)=2/1
=2