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Chapter 12: Q 51. (page 944)

For the partial derivatives $\frac{鈭� f}{鈭� x} = 0$ , find the most general form for a function of two variables $f (x, y)$ , with the given partial derivative.

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Most popular questions from this chapter

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

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

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

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

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

$f (x, y) = \frac{x + y}{x^{2} + y^{2}}, P = (1, 2), v = 鉄� 3, 鈭� 2 鉄�$

Sketch the level curves f(x, y) = c of the following functions for c = 鈭�3, 鈭�2, 鈭�1, 0, 1, 2, and 3:

$f (x, y) = y^{2} 鈭� x^{2}$

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

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For the partial derivatives 鈭�f鈭�x=0, find the most general form for a function of two variables fx,y, with the given partial derivative.

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For the partial derivatives $\frac{鈭� f}{鈭� x} = 0$ , find the most general form for a function of two variables $f (x, y)$ , with the given partial derivative.