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Chapter 12: Q. 32 (page 944)

Find the first-order partial derivatives for the functions in Exercises 27鈥�36.
$g (x, y) = x \cos y$

Short Answer

Expert verified

The first-order partial derivatives are $\frac{鈭� g}{鈭� x} = \cos y and \frac{鈭� g}{鈭� y} = - x \sin y$

Step by step solution

01

Given

The given function is: $g (x, y) = x \cos y$

02

To find

We have to find the first-order partial derivatives.

03

Calculation

$g (x, y) = x \cos y \frac{鈭� g}{鈭� x} = \frac{鈭�}{鈭� x} (x \cos y) \frac{鈭� g}{鈭� x} = \cos y g (x, y) = x \cos y \frac{鈭� g}{鈭� y} = \frac{鈭�}{鈭� y} (x \cos y) \frac{鈭� g}{鈭� y} = x (- \sin y) \frac{鈭� g}{鈭� y} = - x \sin y Hence, \frac{鈭� g}{鈭� x} = \cos y and \frac{鈭� g}{鈭� y} = - x \sin y$

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Most popular questions from this chapter

$Extrema : Find the local maxima, local minima, and saddle points of the given functions f (x, y) = x^{4} + y^{4} + 4 xy$

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$f (x, y, z) = \ln 鈦� (x + y + z)$

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = \tan^{鈭� 1} 鈦� (\frac{y}{x}), P = (鈭� 2, 2)$

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

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

$\lim_{(x, y) 鈫� (4, 蟺)} \tan^{- 1} (\frac{y}{x})$

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