TF. 6 - Calculus

Question

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

Step-by-Step Solution

Answer

Relationship between given partial derivatives is:

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}$

Step 1. Given information

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

Step 2. Finding relationship between ∂ f ∂ s ,   ∂ f ∂ x ,   ∂ f ∂ y ,   ∂ x ∂ s   ,   and   ∂ y ∂ s

Relationship between given partial derivatives is:

$$\begin{gathered} \frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s} \\ LHS = \frac{\partial f}{\partial s}=5s^4 {\cos }^{2}t {\sin }^{3}t \\ RHS=\frac{\partial f}{\partial x} \frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}=\left(2x{y}^3\right)\left(\cos t\right)+\left(3x^2{y}^2\right)\left(\sin t\right) \\ =2\left(s \cos t\right){\left(s \sin t\right)}^3\left(\cos t\right)+3{\left(s \cos t\right)}^2{\left(s \sin t\right)}^2\left(\sin t\right) \\ =2s^4 {\cos }^{2}t {\sin }^{3}t+3s^4 {\cos }^{2}t {\sin }^{3}t \\ =5s^4 {\cos }^{2}t {\sin }^{3}t \end{gathered}$$