求ln√(x²+y²)的导数
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求ln√(x²+y²)的导数
![求ln√(x²+y²)的导数](/uploads/image/z/6447105-9-5.jpg?t=%E6%B1%82ln%E2%88%9A%EF%BC%88x%26%23178%3B%2By%26%23178%3B%EF%BC%89%E7%9A%84%E5%AF%BC%E6%95%B0)
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二元函数.
设Z=ln√(x²+y²)
Z'x=[√(x²+y²)]'x /√(x²+y²)=x/(x²+y²)
Z'y=[√(x²+y²)]'y /√(x²+y²)=y/(x²+y²)
故dZ=x/(x²+y²)dx+y/(x²+y²)dy=[xdx+ydy]/(x²+y²)
一元函数
y=y(x)
[ln√(x²+y²)]'=[ln√(x²+y²)]'x=[√(x²+y²)]'x /√(x²+y²)
=1/2(x²+y²)'x/√(x²+y²) /√(x²+y²)
=1/2[2x+2yy']/(x²+y²)
=(x+yy')/(x²+y²)
二元函数.
设Z=ln√(x²+y²)
Z'x=[√(x²+y²)]'x /√(x²+y²)=x/(x²+y²)
Z'y=[√(x²+y²)]'y /√(x²+y²)=y/(x²+y²)
故dZ=x/(x²+y²)dx+y/(x²+y²)dy=[xdx+ydy]/(x²+y²)
一元函数
y=y(x)
[ln√(x²+y²)]'=[ln√(x²+y²)]'x=[√(x²+y²)]'x /√(x²+y²)
=1/2(x²+y²)'x/√(x²+y²) /√(x²+y²)
=1/2[2x+2yy']/(x²+y²)
=(x+yy')/(x²+y²)