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प्रश्न
W(x, y, z) = xy + yz + zx, x = u – v, y = uv, z = u + v, u, v ∈ R. Find `(del"W")/(del"u"), (del"W")/(del"v")` and evaluate them at `(1/2, 1)`
उत्तर
`(del"W")/(delx) = y + z, (delx)/(del"u") = 1, (delx)/(del"v") = - 1`
`(del"W")/(dely) = x + z, (dely)/(delu) = "v" = (dely)/(delv) = "u"`
`(del"W")/(del"z") = y + x, (delz)/(del"u") = 1, (delz)/(delv) = 1`
`(del"W")/(delu) = (del"W")/(delx) (delx)/(del"u") + (del"W")/(dely) (dely)/(del"u") + (del"W")/(delz) (delz)/(del"u")`
= (y + z) × 1 + (x + z) × v + (y + x) × 1
= uv + u + v + (u – v + u + v) v+ (uv + u – v)
= uv + u + v + uv + uv + uv + u – v
= 4 uv + 2u
`(delw)/(delu) (1/2, 1) = 4 xx 1/2 xx 1 + 2 xx 1/2`
= 2 + 1
= 3
`(del"w")/(delv) = (del"W")/(delx) (delx)/(del"v") + (del"W")/(dely) (dely)/(del"v") + (del"W")/(delz) (delz)/(del"v")`
= (y + z) (– 1) + (x + z) u + (y + x) × 1
= – y – z + xu + zu + y + x
= –u – v + (u – v) u + (u + v) u + u – v
= – u – v + u2 – vu + u2 + vu + u – v
= 2u2 – 2v
`(del"W")/(delv) (1/2, 1) = 2 xx 1/4 - 2 xx 1`
= `1/2 - 2`
= `- 3/2`
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