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Question
Let U(x, y, z) = xyz, x = e–t, y = e–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`
Solution
U(x, y, z) = xyz, x = e–t, y = e–t cos t
`"dU"/("d"x) = yz = "e"^-"t" cos"t" sin"t", ("d"x)/"dt" = - "e"^-"t"`
`"dU"/("d"y) = xz = "e"^-"t" sin"t", ("d"y)/"dt" = - "e"^-"t" cos"t" - "e"^-"t" sin"t"`
`"dU"/("d"z) = xy = "e"^-"t" "e"^-"t" cos"t", ("d"z)/"dt" = cos"t"`
`"dU"/"dt"` = – (e–t cos t sin t) e–t + e–t sin t [e–t (cos t – sin t )] + e–2t cos t (cos t)
= – e–2t cos t sin t – e–2t sin t cos t – e–2t sin²t + e–2t cos²t
= – e–2t (2 sin t cos t + sin2t – cos2t)
= – e–2t [sin 2t – (cos2t – sin2t)]
= – e–2t (sin 2t + cos 2t)
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