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Question
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = log(5x + 3y)
Solution
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`
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