Question

Let U(x, y) = e^x sin y where x = st², y = s²t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1

Answer 1

U(x, y) = e^x sin y where x = st², y = s²t

`(del"U")/(del"s") = (del"U")/(delx) (delx)/(del"s") + (del"U")/(dely) (dely)/(del"s")`

`(del"U")/(delx) = "e"^x siny, (delx)/(del"t") = 2"st", (delx)/(del"s") = "t"^2`

`(del"U")/(dely) = "e"^x cosy, (dely)/(del"t") = "s"^2, (dely)/(del"s") = 2"st"`

`(del"U")/(del"s") = "e"^x siny "t"^2 + "e"^x cosy (2"st")`

= e^_st2 sin (s²t) t² + e^_st2 cos(s²t) 2st

= e^_st2 [t² sin (s²t) + 2st cos (s²t)]

= t e^x [t sin(s²t) + 2s cos (s²t)]

`(del"U")/(del"t")` = e^x sin y 2st + e^x cos y (s²)

= e^_st2 sin(s²t) 2st + e^_st2 cos(s²t) s²

= s e^_st2 [2t sin (s²t) + s cos(s²t)]

At s = t = 1

`(del"U")/(del"s")` e[sin(1) + 2 cos (1)]

= e[sin(1) + 2 cos (1)]

`(del"U")/(del"t")` = e[2 sin(1) + cos (1)]