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Question
Let u(x, y, z) = xy2z3 x = sin t, y = cos t, z = 1 + e2t, Find `"du"/"dt"`
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Solution
u(x, y, z) = xy2z3 x = sin t, y = cos t, z = 1 + e2t
`"du"/("d"x) = y^2z^3 , ("d"x)/"dt"` = cos t
`"du"/("d"y) = 2xyz^3, ("d"y)/"dt"` = – sin t
`"du"/("d"z) = 3xy^2z^2, ("d"z)/"dt"` = 2e2t
`"du"/"dt" = "d"/("d"x) ("d"x)/"dt" + "du"/("d"y) ("d"y)/"dt" + "du"/("d"z) ("d"z)/"dt"`
= y2z3 cos t – 2xyz3 sin t + 6 xy2z2 e2t
= cos2t (1 + e2t)3 cost – 2 (sin t) cost (1 + e2t)3 sin t + 6 (sin t) cos2t) (1 + e2t)2 e2t
= (1 + e2t)2 [cos3t (1 + e2t) – sin t sin2t (1 + e2t) + 6 e2t sin t cos2t]
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