Question

If w(x, y, z) = x² + y² + z², x = e^t, y = e^t sin t and z = e^t cos t, find `("d"w)/"dt"`

Answer 1

w(x, y, z) = x² + y² + z², x = e^t, y = e^t sin t and z = e^t cos t

`("d"w)/"dt" = ("d"w)/("d"x) ("d"x)/"dt" + ("d"w)/("d"y) ("d"y)/"dt" + ("d"w)/"dt" + ("d"w)/("d"z) ("d"z)/"dt"`

`("d"w)/("d"x) = 2x, ("d"x)/"dt" = "e"^"t"`

`("d"w)/("d"y) = 2y, ("d"y)/"dt" = "e"^"t" sin"t" + "e"^"t" cos"t"`

`("d"w)/("d"z) = 2z, ("d"z)/"dt" = "e"^"t" cos"t" - "e"^"t" sin"t"`

`("d"w)/"dt"` = 2x e^t + 2y (e^t sin t + e^t cos t) + 2z (e^t cos t – e^t sin t)

= 2 e^2t + 2 (e^t sin t) (e^t sin t + e^t cos t) + 2 (e^t cos t) (e^t cos t – e^t sin t)

= 2 e^2t [1 + sin²t + sin t cos t + cos²t – sin t cos t]

= 2 e^2t (1 + sin²t + cos²t)  ......[∵ sin²t + cos²t = 1]

= 2 e²t (1 + 1)

= 4 e^2t