Chapter 12: Q. 22 (page 989)

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \tan^{- 1} (\frac{y}{x})$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

$f (x, y) = \tan^{- 1} (\frac{y}{x})$

Step 2. Differentiate

Partially differentiating w.r.t x, y,

$\begin{aligned} \frac{\partial f}{\partial x} = \frac{1}{1 + {(\frac{y}{x})}^{2}} (- \frac{y}{x^{2}}) \\ = - \frac{y}{x^{2} + y^{2}} \end{aligned}$

$\begin{aligned} \frac{\partial f}{\partial y} = \frac{1}{1 + {(\frac{y}{x})}^{2}} (\frac{1}{x}) \\ = \frac{x}{x^{2} + y^{2}} \end{aligned}$

Now, again differentiating w.r.t x and y,

role="math" localid="1649910645072" $\begin{aligned} \frac{\partial^{2} f}{\partial x^{2}} = - \frac{(x^{2} + y^{2}) 0 - y (2 x + 0)}{{(x^{2} + y^{2})}^{2}} \\ = \frac{2 x y}{{(x^{2} + y^{2})}^{2}} \end{aligned}$

Now,

$\begin{aligned} \frac{\partial^{2} f}{\partial y \partial x} = - \frac{(x^{2} + y^{2}) (1) - y (0 + 2 y)}{{(x^{2} + y^{2})}^{2}} \\ = \frac{y^{2} - x^{2}}{{(x^{2} + y^{2})}^{2}} \end{aligned}$

Therefore,

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

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \tan^{- 1} (\frac{y}{x})$

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)=tan−1yx

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

Applied Mathematics

Calculus

Geometry

Decision Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y) = \tan^{- 1} (\frac{y}{x})$