Chapter 12: Q. 20 (page 989)

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$

Short Answer

Expert verified

$\frac{\partial^{2} x}{\partial x^{2}} = \frac{4 x^{4} - 12 x^{2} y^{2}}{{(x^{2} + y^{2})}^{2}}$ and $\frac{\partial^{2} y}{\partial y^{2}} = \frac{8 x^{3} y - 16 x y^{3}}{{(x^{2} + y^{2})}^{3}}$

Step by step solution

Step 1. Given information

$f (x, y) = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$

Step 2. Differentiate

Differentiating partially w.r.t x and y,

$\frac{\partial f}{\partial x} = \frac{(x^{2} + y^{2}) (2 x - 0) - (x^{2} - y^{2}) (2 x + 0)}{{(x^{2} + y^{2})}^{2}}$

localid="1649992615865" $\frac{\partial f}{\partial x} = \frac{4 x y^{2}}{{(x^{2} + y^{2})}^{2}}$

localid="1649992501593">∂f∂y=-4x2y(x2+y2)2

Differentiating $\frac{\partial f}{\partial x} = \frac{4 x y^{2}}{{(x^{2} + y^{2})}^{2}}$ w.r.t x,

localid="1649992959358" $\frac{\partial^{2} x}{\partial x^{2}} = \frac{{(x^{2} + y^{2})}^{2} (4 y^{2}) - {(4 x y^{2})}^{2} (x^{2} + y^{2}) (2 x + 0)}{{(x^{2} + y^{2})}^{4}}$

$\frac{\partial^{2} x}{\partial x^{2}} = \frac{4 x^{2} y^{2} + 4 y^{4} - 16 x^{2} y^{2}}{{(x^{2} + y^{2})}^{4}}$

localid="1649992974376" $\frac{\partial^{2} x}{\partial x^{2}} = \frac{4 x^{4} - 12 x^{2} y^{2}}{{(x^{2} + y^{2})}^{2}}$

Differentiating w.r.t y,

Now,

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Differentiate

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:f(x,y)=x2-y2x2+y2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Differentiate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Probability and Statistics

Mechanics Maths

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$