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Chapter 13: Problem 21

Find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$ z=\frac{x y}{x^{2}+y^{2}} $$

Short Answer

Expert verified
\( \frac{\partial z}{\partial x} = \frac{y^3 - x^2y}{(x^2 + y^2)^2} \) and \( \frac{\partial z}{\partial y} = \frac{x^3 - xy^2}{(x^2 + y^2)^2} \).

Step by step solution

01

Understand the Problem

We are given a function \( z = \frac{xy}{x^2+y^2} \). We need to find the partial derivatives \( \partial z / \partial x \) and \( \partial z / \partial y \). Partial derivatives measure the rate of change of the function with respect to one variable while keeping the other variable constant.
02

Identify the Formula for Partial Derivative

To find a partial derivative of a function \( z = f(x, y) \) with respect to \( x \), treat \( y \) as a constant, and differentiate \( f \) with respect to \( x \). The same principle applies to finding \( \partial z / \partial y \). We will apply these principles by using the quotient rule as \( z \) is a quotient.
03

Apply the Quotient Rule for \( \partial z / \partial x \)

The quotient rule states that for \( u/v \), the derivative \( (uv' - vu')/v^2 \) applies. Let \( u = xy \) and \( v = x^2 + y^2 \). First compute \( u'= y \) (constant) and \( v' = 2x \). Apply the rule:\[ \frac{ \partial z }{ \partial x} = \frac{ (x^2 + y^2)(y) - (xy)(2x) }{ (x^2 + y^2)^2 } \].
04

Simplify \( \partial z / \partial x \)

Simplify the expression:- Numerator: \( y(x^2 + y^2) - 2x^2y = yx^2 + y^3 - 2x^2y = y^3 - x^2y \)- Resulting expression: \[ \frac{ y^3 - x^2y }{ (x^2 + y^2)^2 } \].
05

Apply the Quotient Rule for \( \partial z / \partial y \)

As above, use \( u = xy \) and \( v = x^2 + y^2 \). Compute \( u' = x \) (constant) and \( v' = 2y \):\[ \frac{ \partial z }{ \partial y} = \frac{ (x^2 + y^2)(x) - (xy)(2y) }{ (x^2 + y^2)^2 } \].
06

Simplify \( \partial z / \partial y \)

Simplify the expression:- Numerator: \( x(x^2 + y^2) - 2xy^2 = x^3 + xy^2 - 2xy^2 = x^3 - xy^2 \)- Resulting expression: \[ \frac{ x^3 - xy^2 }{ (x^2 + y^2)^2 } \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule
The quotient rule is a crucial tool in calculus, especially when dealing with functions that are fractions of other functions. This rule provides a method for finding the derivative of a function that is the quotient of two differentiable functions. For a function defined as a quotient, say \( z = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \) and/or \( y \), the quotient rule states that the derivative of \( z \) with respect to one of the variables, say \( x \), is given by:\[ \frac{ \partial z }{ \partial x } = \frac{ u'v - uv' }{ v^2 } \]where \( u' \) is the partial derivative of \( u \) with respect to \( x \), and \( v' \) is the partial derivative of \( v \) with respect to \( x \). It's important to remember that when using the quotient rule, the denominator \( v \) must not be zero to avoid undefined expressions.In the context of the exercise provided, applying the quotient rule requires differentiating both the numerator \( xy \) and the denominator \( x^2 + y^2 \) with respect to the variable of interest. This approach simplifies the process of finding partial derivatives and helps to accurately measure how changes in \( x \) or \( y \) affect the function \( z \).
Rate of Change
Understanding the rate of change in calculus is fundamental, especially when depicting the behavior of functions. In a multivariable setting, such as in this exercise, the idea of rate of change becomes more dynamic since we are dealing with functions of several variables.The partial derivative \( \partial z / \partial x \) measures how the function \( z \) changes as \( x \) changes, while \( y \) remains constant. Similarly, \( \partial z / \partial y \) examines the changes in \( z \) as \( y \) alters, with \( x \) held constant.Using partial derivatives, we quantify the sensitivity of the function \( z \) to changes in its variables. This concept extends beyond just algebraic computations and finds applications in science and engineering鈥攕uch as analyzing how different factors independently influence a system's output. When dealing with rates of change in multivariable calculus, understanding partial derivatives allows us to effectively deconstruct complex systems into individually manageable parts.
Multivariable Calculus
Multivariable calculus deals with functions of two or more variables. It extends the ideas from single-variable calculus to understand higher-dimensional spaces and multiple influences on a system. Involved concepts include partial derivatives, as seen in this exercise. Here, one variable changes while others are held constant, which mirrors real-world scenarios where multiple factors operate in concert. For example, in physics, we might measure how pressure (a function of volume and temperature) changes if we adjust just one of those variables. Moreover, multivariable calculus introduces new tools, like gradients and Lagrange multipliers, expanding our ability to optimize and analyze functions. These tools enable us to find maximum or minimum values for systems with constraints and assess directional change in multidimensional space. Overall, mastering multivariable calculus concepts like those in partial derivatives provides a deeper understanding of how functions behave with multiple inputs, shaping insights required for advanced scientific research and mathematical applications.

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Most popular questions from this chapter

(a) By differentiating implicitly, find the slope of the hyperboloid \(x^{2}+y^{2}-z^{2}=1\) in the \(x\) -direction at the points \((3,4,2 \sqrt{6})\) and \((3,4,-2 \sqrt{6})\) (b) Check the results in part (a) by solving for \(z\) and differentiating the resulting functions directly.

Show that the function satisfies the heat equation $$\frac{\partial z}{\partial t}=c^{2} \frac{\partial^{2} z}{\partial x^{2}} \quad(c>0, \text { constant })$$ $$\text { (a) } z=e^{-t} \sin (x / c) \quad \text { (b) } z=e^{-t} \cos (x / c)$$

A common problem in experimental work is to obtain a mathematical relationship \(y=f(x)\) between two variables \(x\) and \(y\) by "fitting" a curve to points in the plane that correspond to experimentally determined values of \(x\) and \(y,\) say $$ \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) $$ The curve \(y=f(x)\) is called a mathematical model of the data. The general form of the function \(f\) is commonly determined by some underlying physical principle, but sometimes it is just determined by the pattern of the data. We are concerned with fitting a straight line \(y=m x+b\) to data. Usually, the data will not lie on a line (possibly due to experimental error or variations in experimental conditions), so the problem is to find a line that fits the data "best" according to some criterion. One criterion for selecting the line of best fit is to choose \(m\) and \(b\) to minimize the function $$ g(m, b)=\sum_{i=1}^{n}\left(m x_{i}+b-y_{i}\right)^{2} $$ This is called the method of least squares, and the resulting line is called the regression line or the least squares line of best fit. Geometrically, \(\left|m x_{i}+b-y_{i}\right|\) is the vertical distance between the data point \(\left(x_{i}, y_{i}\right)\) and the line \(y=m x+b\) These vertical distances are called the residuals of the data points, so the effect of minimizing \(g(m, b)\) is to minimize the sum of the squares of the residuals. In these exercises, we will derive a formula for the regression line. The purpose of this exercise is to find the values of \(m\) and \(b\) that produce the regression line. (a) To minimize \(g(m, b),\) we start by finding values of \(m\) and \(b\) such that \(\partial g / \partial m=0\) and \(\partial g / \partial b=0 .\) Show that these equations are satisfied if \(m\) and \(b\) satisfy the conditions $$ \left(\sum_{i=1}^{n} x_{i}^{2}\right) m+\left(\sum_{i=1}^{n} x_{i}\right) b=\sum_{i=1}^{n} x_{i} y_{i} $$ \(\left(\sum_{i=1}^{n} x_{i}\right) m+n b=\sum_{i=1}^{n} y_{i}\) (b) Let \(\bar{x}=\left(x_{1}+x_{2}+\cdots+x_{n}\right) / n\) denote the arithmetic average of \(x_{1}, x_{2}, \ldots, x_{n} .\) Use the fact that $$ \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2} \geq 0 $$ to show that $$ n\left(\sum_{i=1}^{n} x_{i}^{2}\right)-\left(\sum_{i=1}^{n} x_{i}\right)^{2} \geq 0 $$ with equality if and only if all the \(x_{i}\) 's are the same. (c) Assuming that not all the \(x_{i}\) 's are the same, prove that the equations in part (a) have the unique solution $$ \begin{aligned} m=& \frac{n \sum_{i=1}^{n} x_{i} y_{i}-\sum_{i=1}^{n} x_{i} \sum_{i=1}^{n} y_{i}}{n \sum_{i=1}^{n} x_{i}^{2}-\left(\sum_{i=1}^{n} x_{i}\right)^{2}} \\ b=& \frac{1}{n}\left(\sum_{i=1}^{n} y_{i}-m \sum_{i=1}^{n} x_{i}\right) \end{aligned} $$ [Note: We have shown that \(g\) has a critical point at these values of \(m\) and \(b\). In the next exercise we will show that \(g\) has an absolute minimum at this critical point. Accepting this to be so, we have shown that the line \(y=m x+b\) is the regression line for these values of \(m \text { and } b .]\)

The area \(A\) of a triangle is given by \(A=\frac{1}{2} a b \sin \theta,\) where \(a\) and \(b\) are the lengths of two sides and \(\theta\) is the angle between these sides. Suppose that \(a=5, b=10,\) and \(\theta=\pi / 3\). (a) Find the rate at which \(A\) changes with respect to \(a\) if \(b\) and \(\theta\) are held constant. (b) Find the rate at which \(A\) changes with respect to \(\theta\) if \(a\) and \(b\) are held constant. (c) Find the rate at which \(b\) changes with respect to \(a\) if \(A\) and \(\theta\) are held constant.

Find \(\partial w / \partial x_{i}\) for \(i=1,2, \ldots, n\) $$ w=\left(\sum_{k=1}^{n} x_{k}\right)^{1 / n} $$
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