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Chapter 12: Problem 45

Find the first partial derivatives of the following functions. $$f(x, y, z)=x y+x z+y z$$

Short Answer

Expert verified
Answer: The first partial derivatives are: 1. With respect to \(x\): \(\frac{\partial f}{\partial x} = y + z\) 2. With respect to \(y\): \(\frac{\partial f}{\partial y} = x + z\) 3. With respect to \(z\): \(\frac{\partial f}{\partial z} = x + y\)

Step by step solution

01

Finding the partial derivative with respect to x

First, we need to find the partial derivative of the function with respect to \(x\). To do this, we treat \(y\) and \(z\) as constants and take the derivative. The partial derivative with respect to \(x\) is: $$\frac{\partial}{\partial x}(xy + xz + yz) = y + z$$
02

Finding the partial derivative with respect to y

Next, we need to find the partial derivative of the function with respect to \(y\). To do this, we treat \(x\) and \(z\) as constants and take the derivative. The partial derivative with respect to \(y\) is: $$\frac{\partial}{\partial y}(xy + xz + yz) = x + z$$
03

Finding the partial derivative with respect to z

Finally, we need to find the partial derivative of the function with respect to \(z\). To do this, we treat \(x\) and \(y\) as constants and take the derivative. The partial derivative with respect to \(z\) is: $$\frac{\partial}{\partial z}(xy + xz + yz) = x + y$$ So, the first partial derivatives of the function are: $$\frac{\partial f}{\partial x} = y + z$$ $$\frac{\partial f}{\partial y} = x + z$$ $$\frac{\partial f}{\partial z} = x + y$$

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Key Concepts

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

Multivariable Calculus
When venturing into the realm of calculus, multivariable calculus is a thrilling expansion that deals with functions of more than one variable. Unlike in single-variable calculus, where you deal with functions involving one variable like f(x), multivariable calculus involves functions like f(x, y, z), which depend on several variables.

Consider it as a progression from studying a line (1D) to exploring the landscapes of surfaces (2D) and even higher dimensions (3D and beyond). It provides a powerful toolkit for understanding how these multi-dimensional functions change and interconnect. This field includes concepts like partial derivatives, multiple integrals, and vector calculus. It is essential in physics, engineering, economics, and even in understanding the geometry of higher-dimensional spaces.
First Partial Derivative
The first partial derivative is a foundational concept in multivariable calculus. To understand them, think about taking a regular derivative, but with a twist. You treat every variable except the one you鈥檙e focusing on as a constant.

For instance, when finding the first partial derivative with respect to x for a function f(x, y), you simply ignore the influence of y as if it were a fixed number. Doing so reveals how the function's value changes with slight changes in x, holding y constant. This is fundamental for examining the slopes and rates of change in multi-dimensional settings. It鈥檚 like asking how steep a hill is if you're moving eastward but staying on the same northern line.
Derivatives of Functions
In the broader picture, derivatives of functions are the bread and butter of calculus. They measure the instantaneous rate of change of a function concerning one of its variables. You've probably encountered a simple derivative like f'(x) in single-variable calculus, telling you how fast f(x) is changing at any point x.

In multivariable functions, each first partial derivative tells you about changes in a specific direction. Combining these derivates can provide a complete picture of how a function behaves locally. It's like understanding all the different ways you can climb a mountain 鈥� east, north, or at an angle. Each partial derivative gives you information about the ascent in each independent direction, crafting a detailed understanding of the topography.

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Most popular questions from this chapter

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\) (Hint: Find the point \(P\) on the plane closest to \(P_{0}\).)

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

Determine whether the following statements are true and give an explanation or counterexample. a. The plane passing through the point (1,1,1) with a normal vector \(\mathbf{n}=\langle 1,2,-3\rangle\) is the same as the plane passing through the point (3,0,1) with a normal vector \(\mathbf{n}=\langle-2,-4,6\rangle\) b. The equations \(x+y-z=1\) and \(-x-y+z=1\) describe the same plane. c. Given a plane \(Q\), there is exactly one plane orthogonal to \(Q\). d. Given a line \(\ell\) and a point \(P_{0}\) not on \(\ell\), there is exactly one plane that contains \(\ell\) and passes through \(P_{0}\). e. Given a plane \(R\) and a point \(P_{0}\), there is exactly one plane that is orthogonal to \(R\) and passes through \(P_{0}\) f. Any two distinct lines in \(\mathbb{R}^{3}\) determine a unique plane. g. If plane \(Q\) is orthogonal to plane \(R\) and plane \(R\) is orthogonal to plane \(S\), then plane \(Q\) is orthogonal to plane \(S\).

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)

Identify and briefly describe the surfaces defined by the following equations. $$x^{2} / 4+y^{2}-2 x-10 y-z^{2}+41=0$$
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