高数题
g(x,y)=f(y/x)+yf(x/y)
∂g/∂x=f'(y/x)*(-y/x^2)+yf'(x/y)*(1/y)=f'(y/x)*(-y/x^2)+f'(x/y)
∂^2g/∂x^2=[f''(y/x)*(-y/x^2)*(-y/x^2)+f'(y/x)*(2y/x^3)]+yf''(x/y)*(1/y)*(1/y)=f''(y/x)*(y^2/x^4)+f'(y/x)*(2y/x^3)+f''(x/y)*(1/y)
∂g/∂y=f'(y/x)*(1/x)+f(x/y)+yf'(x/y)*(-x/y^2)=f'(...全部
g(x,y)=f(y/x)+yf(x/y)
∂g/∂x=f'(y/x)*(-y/x^2)+yf'(x/y)*(1/y)=f'(y/x)*(-y/x^2)+f'(x/y)
∂^2g/∂x^2=[f''(y/x)*(-y/x^2)*(-y/x^2)+f'(y/x)*(2y/x^3)]+yf''(x/y)*(1/y)*(1/y)=f''(y/x)*(y^2/x^4)+f'(y/x)*(2y/x^3)+f''(x/y)*(1/y)
∂g/∂y=f'(y/x)*(1/x)+f(x/y)+yf'(x/y)*(-x/y^2)=f'(y/x)*(1/x)+f(x/y)+f'(x/y)*(-x/y)
∂^2g/∂y^2=f''(y/x)*(1/x)*(1/x)+f'(x/y)*(-x/y^2)+f''(x/y)*(-x/y^2)*(-x/y)+f'(x/y)*(x/y^2)=f''(y/x)*(1/x^2)+f''(x/y)*(x^2/y^3)
所以x^2∂^2g/∂x^2-y^2∂^2g/∂y^2=f'(x/y)(2y/x^2)。
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