Question

Let u(x, y, z) = xy²z³ x = sin t, y = cos t, z = 1 + e^2t, Find `"du"/"dt"`

Answer 1

u(x, y, z) = xy²z³ x = sin t, y = cos t, z = 1 + e^2t 

`"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"` = 2e^2t

`"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 e^2t

= cos²t (1 + e^2t)³ cost – 2 (sin t) cost (1 + e^2t)³ sin t + 6 (sin t) cos²t) (1 + e^2t)² e^2t

= (1 + e^2t)² [cos³t (1 + e^2t) – sin t sin2t (1 + e^2t) + 6 e^2t sin t cos²t]