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Chapter 12: TF. 3 (page 946)

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

Short Answer

Expert verified

$\frac{鈭� y}{鈭� s} = \sin t \frac{鈭� y}{鈭� t} = s \cos t$

Step by step solution

Step 1. Given information

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

Step 2. Finding partial derivatives

$y = s \sin t . 鈬� \frac{鈭� y}{鈭� s} = \sin t 鈬� \frac{鈭� y}{鈭� t} = s \cos t$

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Most popular questions from this chapter

$Let z = (f {(x, y))}^{n} . Show that \frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} and \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} .$

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

$\lim_{(x, y, z) 鈫� (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}}$

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

$\frac{鈭� z}{鈭� 胃} w h e n z = (x^{2} + x y) e^{y}, x = r \cos 胃 a n d y = r \sin 胃$

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.

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

How do you find the critical points of a function of two variables, $f (x, y)$ ? What is the significance of the critical points?

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Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Find鈭�y鈭�sand鈭�y鈭�t.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding partial derivatives

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