Question

x方ye的x+y次方偏导数

Answer 1

问：x方ye的x+y次方偏导数

答：两边取对数：lnz = xy lnx∂lnz/∂x = ∂z/z∂x = y(lnx+1)∂z/∂x = z(lnx+1) = x^(xy) (lnx+1)y∂lnz/∂y = ∂z/z∂y = xlnx∂z/∂y = zxlnx = x^(xy+1) lnx

答：z=x^(x^y)lnz=(x^y) lnx∂lnz/∂x=yx^(y-1) lnx + x^y/x = yx^(y-1) lnx + x^(y-1) = x^(y-1)(ylnx+1)∂lnz/∂x = (1/z)∂z/∂x = x^(y-1)(ylnx+1)∂z/∂x = x^(y-1)(ylnx+1)x^(x^y) (1)∂lnz/∂y = (1/z)∂z/∂y = lnx(x^y)lnx∂z/∂y = lnx(x^y)lnx x^(x^y) = ln²x [x^(x^y+y)] (2)