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第八题,高数,多元函数微分学,多元复合函数求导法则,,求讲解
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t^(k)*f(x,y,z)=f(tx,ty,tz) 对t求导数有:
k*t^(k-1)*f(x,y,z)=x*df/dtx+y*df/dty+z*df/dtz①
t^(k)*f(x,y,z)=f(tx,ty,tz) 对x求导数有:
t^(k)*df/dx=t*df/dtx ∴ t^(k-1)*df/dx= df/dtx
同理:t^(k-1)*df/dy= df/dty ; t^(k-1)*df/dz= df/dtz 代入①
k*t^(k-1)*f(x,y,z)=x*t^(k-1)*df/dx+y*t^(k-1)*df/dy+ z*t^(k-1)*df/dz
∴k*f(x,y,z)=x*df/dx+y*df/dy+ z*df/dz
k*t^(k-1)*f(x,y,z)=x*df/dtx+y*df/dty+z*df/dtz①
t^(k)*f(x,y,z)=f(tx,ty,tz) 对x求导数有:
t^(k)*df/dx=t*df/dtx ∴ t^(k-1)*df/dx= df/dtx
同理:t^(k-1)*df/dy= df/dty ; t^(k-1)*df/dz= df/dtz 代入①
k*t^(k-1)*f(x,y,z)=x*t^(k-1)*df/dx+y*t^(k-1)*df/dy+ z*t^(k-1)*df/dz
∴k*f(x,y,z)=x*df/dx+y*df/dy+ z*df/dz
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