Question

大一的高等数学题目 以下的3、4题不会做，请指教详细做法，谢谢

Answer 1

3. z=arctan[(x+y)/(x-y)]
dz=[(x-y)(dx+dy)-(x+y)(dx-dy)]/{(x-y)^2[1+(x+y)^2/(x-y)^2]}
=(-2ydx+2xdy)/[(x-y)^2+(x+y)^2]
=(-ydx+xdy)/(x^2+y^2)
4. z=yf(y/x)+xg(x/y)
偏z/偏x=yf'*(-y/x^2)+g+xg'*1/y=-y^2/x^2f'+x/yg'+g
偏^2(z)/偏x^2=-y^2/x^2f''(-y/x^2)+2y^2/x^3f'+x/yg''(1/y)+1/yg'+g'(1/y)
=y^3/x^4*f''+x/y^2g''+2y^2/x^3f'+2/yg'
偏^2(z)/偏x偏y=-y^2/x^3f''-2y/x^2f'-x^2/y^3g''-x/y^2g'-x/y^2g'
=-y^2/x^3f''-x^2/y^3g''-2y/x^2f'-2x/y^2g'
x偏^2z/偏x^2+y偏^2(z)/偏x偏y
=y^3/x^3*f''+x^2/y^2g''+2y^2/x^2f'+2x/yg' -y^3/x^3f''-x^2/y^2g''-2y^2/x^2f'-2x/yg'
=0

Answer 2

看不清 希望LZ把题目打下来