Advertisements
Advertisements
Question
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)`
Solution
`(del"G")/(delx) = "e"^(x + 3y) [(2x)/(x^2 + y^2)] + log (x^2 + y^2) ["e"^(x + 3y)]`
= `"e"^(x + 3y) [(2x)/(x^2 + y^2) + log(x^2 + y^2)]`
`(del"G")/(dely) = "e"^(x + 3y) [(2x)/(x^2 + y^2)] + log (x^2 + y^2) [3"e"^(x + 3y)]`
`(del"G")/(dely) = "e"^(x + 3y) [(2y)/(x^2 + y^2) + 3log(x^2 + y^2)]`
At (– 1, 1)
`(del"G")/(delx) = "e"^(-1 + 3) [(-2)/(1 + 1) + log(1 + 1)]`
= `"e"^2[-1 + log2]`
= `"e"^2 [log2 - 1]`
`(del"G")/(dely) = "e"^(-1 + 3) [2/(1 + 1) + 3log(1 + 1)]`
= `"e"^2[1 + 3 log2]`
= `"e"^2[1 + log 8]`
APPEARS IN
RELATED QUESTIONS
If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.
If u = exy, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x2 + y2).
Let u = x cos y + y cos x. Verify `(del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")`
If u = 4x2 + 4xy + y2 + 4x + 32y + 16, then `(del^2"u")/(del"y" del"x")` is equal to:
If u = `e^(x^2)` then `(del"u")/(delx)` is equal to:
Find the partial dervatives of the following functions at indicated points.
g(x, y) = 3x2 + y2 + 5x + 2, (2, – 5)
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `(3x)/(y + sinx)`
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) = log(5x + 3y)
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 V(x, y) = ex (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^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)`
If v(x, y) = `x^2 - xy + 1/4 y^2 + 7, x, y ∈ "R"`, find the differential dv
Let V (x, y, z) = xy + yz + zx, x, y, z ∈ 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
