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Chapter 12: TF. 6 (page 946)

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

Short Answer

Expert verified

Relationship between given partial derivatives is:

$\frac{鈭� f}{鈭� s} = \frac{鈭� f}{鈭� x} \frac{鈭� x}{鈭� s} + \frac{鈭� f}{鈭� y} \frac{鈭� y}{鈭� s}$

Step by step solution

01

Step 1. Given information

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

02

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

Relationship between given partial derivatives is:

$\frac{鈭� f}{鈭� s} = \frac{鈭� f}{鈭� x} \frac{鈭� x}{鈭� s} + \frac{鈭� f}{鈭� y} \frac{鈭� y}{鈭� s} L H S = \frac{鈭� f}{鈭� s} = 5 s^{4} \cos^{2} t \sin^{3} t R H S = \frac{鈭� f}{鈭� x} \frac{鈭� x}{鈭� s} + \frac{鈭� f}{鈭� y} \frac{鈭� y}{鈭� s} = (2 x y^{3}) (\cos t) + (3 x^{2} y^{2}) (\sin t) = 2 (s \cos t) {(s \sin t)}^{3} (\cos t) + 3 {(s \cos t)}^{2} {(s \sin t)}^{2} (\sin t) = 2 s^{4} \cos^{2} t \sin^{3} t + 3 s^{4} \cos^{2} t \sin^{3} t = 5 s^{4} \cos^{2} t \sin^{3} t$

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Most popular questions from this chapter

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f (x, y, z) = x + y + z when x^{2} + 4 y^{2} + 16 z^{2} = 64$

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x + y}{x^{2} + y^{2}}$

Solve the exact differential equations in Exercises 63鈥�66. $y \cos (x y) 鈭� 3 + (x \cos (x y) + 2) \frac{d y}{d x} = 0 .$

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable. $\frac{d x}{d t} w h e n x = r \cos 胃, r = t^{2} 鈭� 5, a n d 胃 = t^{3} + 1$

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