Advertisements
Advertisements
Question
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = log(5x + 3y)
Advertisements
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`
APPEARS IN
RELATED QUESTIONS
If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.
Verify Euler’s theorem for the function u = x3 + y3 + 3xy2.
If u = x3 + 3xy2 + y3 then `(del^2"u")/(del "y" del x)`is:
Find the partial derivatives of the following functions at indicated points.
h(x, y, z) = x sin (xy) + z2x, `(2, pi/4, 1)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `(3x)/(y + sinx)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `tan^-1 (x/y)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `cos(x^2 - 3xy)`
If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`
If U(x, y, z) = `log(x^3 + y^3 + z^3)`, find `(del"U")/(delx) + (del"U")/(dely) + (del"U")/(del"z)`
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = xey + 3x2y
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = x2 + 3xy – 7y + cos(5x)
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)`
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)
Let z(x, y) = x2y + 3xy4, x, y ∈ R, Find the linear approximation for z at (2, –1)
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
