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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)`
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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]`
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