TF. 4 - Calculus

Question

$$\begin{gathered} \mathrm{A} \mathrm{chain} \mathrm{rule} \mathrm{for} \mathrm{functions} \mathrm{of} \mathrm{two} \mathrm{variables}: \mathrm{Let} f(x, y)=x^2{y}^3, x=s \cos t, \mathrm{and} y=s \sin t. \\ \mathrm{Substitute} \mathrm{the} \mathrm{expressions} \mathrm{for} x \mathrm{and} y \mathrm{into} f(x, y). \end{gathered}$$

Step-by-Step Solution

Answer

$f(x, y)=s^5 {\cos }^2 t {\sin }^3 t$

Step 1. Given information

$f(x, y)=x^2{y}^3, x=s \cos t, \mathrm{and} y=s \sin t.$

Step 2. Substitute   the   expressions   for   x   and   y   into   f ( x ,   y )

$$\begin{gathered} f(x, y)=x^2{y}^3, x=s \cos t, \mathrm{and} y=s \sin t \\ \Rightarrow f(x, y)={(s \cos t)}^2\times {(s \sin t)}^3 \\ \Rightarrow f(x, y)=(s^2 {\cos }^2 t)\times (s^3 {\sin }^3 t) \\ \Rightarrow f(x, y)=s^5 {\cos }^2 t {\sin }^3 t \end{gathered}$$