TF. 1 - Calculus

Question

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

Step-by-Step Solution

Answer

$$\begin{gathered} \frac{\partial f}{\partial x}=2x{y}^3 \\ \frac{\partial f}{\partial y}=3x^2{y}^2 \end{gathered}$$

Step 1. Given information

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

Step 2. Finding partial derivatives

$$\begin{gathered} f(x, y)=x^2{y}^3 \\ \Rightarrow \frac{\partial f}{\partial x}=2x{y}^3 \\ \Rightarrow \frac{\partial f}{\partial y}=3x^2{y}^2 \end{gathered}$$