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Question
Let w(x, y) = 6x3 – 3xy + 2y2, x = es, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0
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Solution
w(x, y) = 6x3 – 3xy + 2y2
`("d"w)/"ds" = ("d"w)/("d"x) ("d"x)/"ds" + ("d"w)/("d"y) ("d"y)/"ds"`
`("d"w)/("d"x) = 18x^2 - 3y, ("d"x)/"ds" = "e"^"s"`
`("d"w)/("d"y) = - 3x + 4y, ("d"y)/"ds" = - sin "s"`
`("d"w)/"ds" = (18"e"^(2"s") - 3 cos "s")"e"^"s" + (- 3"e"^(2"s") + 4cos "s")(- sin "s")`
`("d"w)/"ds" = 18"e"^(3"s") - 3"e"^"s" cos "s" + 3"e"^"s" sin "s" - 4 cos "s" sin "s"`
At s = 0,
`("d"w)/"ds" = 18"e"^circ - "e"^circ cos 0 + "e"^circ sin 0 - 4 cos 0 sin 0`
= 18 – 3 + 0 + 0
`("d"w)/"ds"` = 15
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