Question

Let z(x, y) = x³ – 3x²y³ where x = se^t, y = se^–t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(delt)`

Answer 1

z(x, y) = x³ – 3x²y³ 

`(delx)/(dels) = "e"^"t", (delx)/(del"t") = "se"^"t"`

`(dely)/(del"s") = "e"^-"t", (dely)/(del"t") = - "se"^-"t"`

`(delz)/(delx) = 3x^2 - 6x  y^3`

`(delz)/(dely) = - 9x^2 y^2`

`(delz)/(del"s") = (delz)/(delx) (delx)/(dels) + (delz)/(dely) (dely)/(del"s")`

(3x² – 6xy³) e^t – 9 (xy)² e^–t

[3 (s e^t)² – 6 (s e^t) (se^–t)³]e^t

– 9 (s e^t s e^–t)² × e^–t

= (3 s² e^2t – 6 s⁴ e^–2t) e^t – 9 s⁴ e^–t

= 3s² [(e^2t – 2s² e^–2t) e^t – 3 e^–t s²]

`(delz)/(del"s")` = 3s² e^t (e^2t – 2s² e^–2t – 3 e^–2t s²)

`(delz)/(del"t") = (delz)/(delx) (delx)/(del"t") + (delz)/(dely) (dely)/(del"t")`

= (3x² – 6xy³) (s e^t)+ (– 9x²y²) (– s e^–t)

= [3 (s e^t)² – 6 (s e^t) (s e^–t)³] s e^t + 9(se^t s e^–t)² s e^–t

= (3 s² e^2t – 6 s⁴ e^–2t) s e^t + 9 s⁵ e^–t

= 3 s³ e^3t – 6 s⁵ e^–t + 9 s⁵ e^–t

= 3 s³ e^3t + 3 s⁵ e^–t 

`(dely)/(del"t")` = 3 s³ (e^3t + s² e^–t)