Xy-sin(x+y)=1 求Y的导数

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Xy-sin(x+y)=1求Y的导数Xy-sin(x+y)=1求Y的导数Xy-sin(x+y)=1求Y的导数对x求导(xy)''=x''*y+x*y''=y+x*y''[sin(x+y)]''=cos(x+y)

Xy-sin(x+y)=1 求Y的导数
Xy-sin(x+y)=1 求Y的导数

Xy-sin(x+y)=1 求Y的导数
对x求导
(xy)'=x'*y+x*y'=y+x*y'
[sin(x+y)]'=cos(x+y)*(x+y)'=(1+y')cos(x+y)=cos(x+y)+cos(x+y)*y'
1'=0
所以y+x*y'-cos(x+y)-cos(x+y)*y'=0
y'=[y-cos(x+y)]/[cos(x+y)-x]

等式两边同时对x 求导
y+x*Y'- Cos(x+y)* (1+Y')=0
所以 Y'= (y-cos(x+y)) / (cos(x+y)-x)
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隐函数求导，两边同时对X求导
得到
(y+xy^)-cos(x+y)*(x+y)^=0
(y+xy^)-cos(x+y)*(1+y^)=0
y^=(y-cos(x+y))/(cos(x+y)-x)
用到相乘函数的求导公式以及复合函数的求导公式

Xy-sin(x+y)=1两边同时求导数，可得到：
(y+xy')-cos(x+y)(x+y)'=0
y+xy'-cos(x+y)(1+y')=0
y+xy'-cos(x+y)-y'cos(x+y)=0
y'(x-cos(x+y))=cos(x+y)-y
所以：
y'=[cos(x+y)-y]/[x-cos(x+y)].

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