If `z =log(e^x+e^y) "show that rt" - s^2 = 0 "where r"= (del^2z)/(delx^2),t=(del^2z)/(dely^2)"s"=(del^2z)/(delx dely)`
Solution 1
`z =log(e^x+e^y)`
(1)`(delz)/(delx)=e^x/(e^x+e^y)`
`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`
`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2` .............(1)
(2) `(delz)/(dely)=e^y/(e^x+e^y)`
`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`
t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2` ................(2)
Solution 2
`z =log(e^x+e^y)`
(1)`(delz)/(delx)=e^x/(e^x+e^y)`
`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`
`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2` .............(1)
(2) `(delz)/(dely)=e^y/(e^x+e^y)`
`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`
t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2` ................(2)
(3) `(delz)/(delx)=e^x/(e^x+e^y)^`
s `=(del^2z)/(delxdely)=e^(xy)/(e^x+e^y)^2` ................(3)
From (1), (2) and (3) we get,
rt `=(e^(xy)/(e^x+e^y)^2)xx(e^(xy)/(e^x+e^y)^2)=(e^(xy)/(e^x+e^y)^2)^2=(e^(2xy)/(e^x+e^y)^2)` .............(4)
s2 = `(e^(xy)/(e^x+e^y)^2)^2= (e^(2xy)/(e^x+e^y)^2)`................(5)
From (4) and (5) we get,
rt-s2 = `(e^(2xy)/(e^x+e^y)^2)-(e^(2xy)/(e^x+e^y)^2)=0`
Hence proved rt - `s^2`= 0
Solution 3
`z =log(e^x+e^y)`
(1)`(delz)/(delx)=e^x/(e^x+e^y)`
`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`
`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2` .............(1)
(2) `(delz)/(dely)=e^y/(e^x+e^y)`
`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`
t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2` ................(2)
(3) `(delz)/(delx)=e^x/(e^x+e^y)^`
s `=(del^2z)/(delxdely)=e^(xy)/(e^x+e^y)^2` ................(3)
From (1), (2) and (3) we get,
rt `=(e^(xy)/(e^x+e^y)^2)xx(e^(xy)/(e^x+e^y)^2)=(e^(xy)/(e^x+e^y)^2)^2=(e^(2xy)/(e^x+e^y)^2)` .............(4)
s2 = `(e^(xy)/(e^x+e^y)^2)^2= (e^(2xy)/(e^x+e^y)^2)`................(5)
From (4) and (5) we get,
rt-s2 = `(e^(2xy)/(e^x+e^y)^2)-(e^(2xy)/(e^x+e^y)^2)=0`
Hence proved rt - `s^2`= 0
