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Chapter 12: Q. 19 (page 989)

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

Short Answer

Expert verified

$\frac{\partial^{2} f}{\partial x^{2}} = \frac{2 x^{3} + 6 x^{2} y - 6 x y^{2} - 2 y^{3}}{{(x^{2} + y^{2})}^{2}}$ and $\frac{\partial^{2} f}{\partial y^{2}} = \frac{2 x^{3} - 6 x^{2} y + 6 x y^{2} - 2 y^{3}}{{(x^{2} + y^{2})}^{3}}$

Step by step solution

Step 1. Given information

Step 2. Differentiate

Partially differentiating w.r.t x and y,

Now,

and,

Therefore,

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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.

$f (x, y, z) = x y z when x^{2} + y^{2} + z^{2} = 9$

Fill in the blanks to complete the limit rules. You may assume that $lim_{x \to a} f (x)$ and $lim_{x \to a} g (x)$ exists and that k is a scalar.

$lim_{x \to a} (f (x) + g (x)) =$

Given a function of three variables, $w = f (x, y, z),$ and a constraint equation $g (x, y, z) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

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

Describe the meanings of each of the following mathematical expressions:

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

Short Answer

Step by step solution

Step 1. Given information

Step 2. Differentiate

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Most popular questions from this chapter

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