Question

For the following functions find the gxy, gxx, gyyand gyx

g(x, y) = log(5x + 3y)

Answer 1

gx = $(del"g")/(delx) = 1/(5x + 3y) (5) = 5/(5x + 3y)$

gy = $(del"g")/(dely) = 1/(5x + 3y) (3) = 3/(5x + 3y)$

gxx = $(del^2"g")/(delx^2)$

= $del/(delx) [(delg)/(delx)]$

= $del/(delx) [5/(5x + 3y)]$

= $((5x + 3y)(0) - 5(5))/(5x + 3y)^2$

= $(- 25)/(5x + 3y)^2$

gyy = $(del^2"g")/(dely^2)$

= $del/(dely) [(del"g")/(dely)]$

= $del/(dely) [3/(5x + 3y)]$

= $((5x + 3y)(0) - 3(3))/(5x + 3y)^2$

= $(- 9)/(5x + 3y)^2$

gxy = $(del^2"g")/(delxdely)$

= $del/(delx) [(del"g")/(dely)]$

= $del/(delx) [3/(5x + 3y)]$

= $(- 3)/(5x + 3y)^2 (5)$

= $(- 15)/(5x + 3y)^2$

gyx = $(del^2"g")/(delydelx)$

= $del/(dely) [(del"g")/(delx)]$

= $del/(dely) [5/(5x + 3y)]$

= $(- 5)/(5x + 3y)^2 (3)$

= $(- 15)/(5x + 3y)^2$