Advertisements
Advertisements
Question
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = xey + 3x2y
Solution
`(del"g")/(del"y") = "g"_x = "e"^y + 6xy`
`(del"g")/(dely) = "g"_y = x"e"^y + 3x^2`
gxx = `(del^2"g")/(delx^2)`
= `del/(delx) [(del"g")/(delx)]`
= `del/(delx) ["e"^y + 6xy]`
= 0 + 6y
= 6y
gyy = `(del^2"g")/(dely^2)`
= `del/(dely) [(del"g")/(dely)]`
= `del/(dely) [x"e"^y + 3x^2]`
= `x"e"^y`
gxy = `(del^2"g")/(delxdely)`
= `del/(delx) [(del"g")/(dely)]`
= `del/(delx) [x"e"^y + 3x^2]`
= `"e"^y + 6x`
gyx = `(del^2"g")/(delydelx)`
= `del/(dely) [(del"g")/(delx)]`
= `del/(dely) ["e"^y + 6xy]`
= `"e"^y + 6x`
APPEARS IN
RELATED QUESTIONS
Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`
Let u = `log (x^4 - y^4)/(x - y).` Using Euler’s theorem show that `x (del"u")/(del"x") + y(del"u")/(del"y")` = 3.
If u = x3 + 3xy2 + y3 then `(del^2"u")/(del "y" del x)`is:
If u = `e^(x^2)` then `(del"u")/(delx)` is equal to:
Find the partial dervatives of the following functions at indicated points.
f(x, y) = 3x2 – 2xy + y2 + 5x + 2, (2, – 5)
Find the partial dervatives of the following functions at indicated points.
g(x, y) = 3x2 + y2 + 5x + 2, (2, – 5)
Find the partial derivatives of the following functions at indicated points.
h(x, y, z) = x sin (xy) + z2x, `(2, pi/4, 1)`
Find the partial derivatives of the following functions at the indicated points.
`"G"(x, y) = "e"^(x + 3y) log(x^2 + y^2), (- 1, 1)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `cos(x^2 - 3xy)`
Let w(x, y, z) = `1/sqrt(x^2 + y^2 + z^2)` = 1, (x, y, z) ≠ (0, 0, 0), show that `(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2)` = 0
If w(x, y) = xy + sin(xy), then Prove that `(del^2w)/(delydelx) = (del^2w)/(delxdely)`
If v(x, y, z) = x3 + y3 + z3 + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C (x, y) = 8x + 6y + 2000 respectively. Find the profit function P(x, y)
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively. Find `(del"P")/(delx)` (1200, 1800) and `(del"P")/(dely)` (1200, 1800) and interpret these results
If v(x, y) = `x^2 - xy + 1/4 y^2 + 7, x, y ∈ "R"`, find the differential dv
Choose the correct alternative:
If g(x, y) = 3x2 – 5y + 2y2, x(t) = et and y(t) = cos t then `"dg"/"dt"` is equal to
