Problem 21
Question
Find the first partial derivatives of the function. \(f(x, y, z)=x y z+x y^{2}+y z^{2}+z x^{2}\)
Step-by-Step Solution
Verified Answer
The first partial derivatives of the function \(f(x, y, z) = xyz + xy^2 + yz^2 + zx^2\) are:
1. \(\frac{\partial f}{\partial x} = yz + 2xz\)
2. \(\frac{\partial f}{\partial y} = xz + 2xy + z^2\)
3. \(\frac{\partial f}{\partial z} = xy + 2yz + x^2\)
1Step 1: Find the partial derivative with respect to x
To find \(\frac{\partial f}{\partial x}\), take the derivative of the given function f(x, y, z) with respect to x, treating y and z as constants.
Given function:
\(f(x,y,z)=xyz+xy^2+yz^2+zx^2\)
Now, find the partial derivative with respect to x:
\(\frac{\partial f}{\partial x}=(y*z)+(0)+(0)+(z*2x)\)
So,
\(\frac{\partial f}{\partial x}=yz+2xz\).
2Step 2: Find the partial derivative with respect to y
To find \(\frac{\partial f}{\partial y}\), take the derivative of the given function f(x, y, z) with respect to y, treating x and z as constants.
Given function:
\(f(x,y,z)=xyz+xy^2+yz^2+zx^2\)
Now, find the partial derivative with respect to y:
\(\frac{\partial f}{\partial y}=(x*z)+(x*2y)+(z^2*1)+(0)\)
So,
\(\frac{\partial f}{\partial y}=xz+2xy+z^2\).
3Step 3: Find the partial derivative with respect to z
To find \(\frac{\partial f}{\partial z}\), take the derivative of the given function f(x, y, z) with respect to z, treating x and y as constants.
Given function:
\(f(x,y,z)=xyz+xy^2+yz^2+zx^2\)
Now, find the partial derivative with respect to z:
\(\frac{\partial f}{\partial z}=(x*y)+(0)+(y*2z)+(x^2*1)\)
So,
\(\frac{\partial f}{\partial z}=xy+2yz+x^2\).
As a result, we found the first partial derivatives of the function \(f(x, y, z)\) is:
1. \(\frac{\partial f}{\partial x} = yz+2xz\)
2. \(\frac{\partial f}{\partial y} = xz+2xy+z^2\)
3. \(\frac{\partial f}{\partial z} = xy+2yz+x^2\)
Key Concepts
Multivariable CalculusFirst Partial DerivativePartial Differentiation
Multivariable Calculus
Multivariable calculus, also referred to as multivariate calculus, is an extension of single-variable calculus to functions of several variables. This field of mathematics provides tools for analyzing functions that depend on more than one variable, such as the case with the function
\(f(x, y, z) = xyz + xy^2 + yz^2 + zx^2\).
In many real-world scenarios, it's crucial to understand how these multivariate functions behave, particularly how they change in response to changes in any one of the variables. This is where the concept of partial derivatives, which we explore in the subsequent sections, becomes core to multivariable calculus. From optimizing functions in economics to modelling physical systems in engineering, multivariable calculus plays an imperative role in many scientific fields.
\(f(x, y, z) = xyz + xy^2 + yz^2 + zx^2\).
In many real-world scenarios, it's crucial to understand how these multivariate functions behave, particularly how they change in response to changes in any one of the variables. This is where the concept of partial derivatives, which we explore in the subsequent sections, becomes core to multivariable calculus. From optimizing functions in economics to modelling physical systems in engineering, multivariable calculus plays an imperative role in many scientific fields.
First Partial Derivative
The first partial derivative is a fundamental concept in multivariable calculus. It represents the rate at which a function changes as one of its variables is varied, while the other variables are held constant. Consider our function of three variables
\(f(x, y, z)\).
To understand how this function changes in the 'x' direction alone, we calculate the first partial derivative with respect to 'x'.
The result,
\(\frac{\/partial f}{\partial x} = yz + 2xz\)
, tells us how the function's value shifts per unit change in 'x' when 'y' and 'z' stay fixed. These computations are akin to taking the derivative of a single-variable function in standard calculus. However, for multivariable functions, we perform this process for each variable in turn, which yields a set of partial derivatives—one for each variable, providing a broader picture of the function's behavior.
\(f(x, y, z)\).
To understand how this function changes in the 'x' direction alone, we calculate the first partial derivative with respect to 'x'.
The result,
\(\frac{\/partial f}{\partial x} = yz + 2xz\)
, tells us how the function's value shifts per unit change in 'x' when 'y' and 'z' stay fixed. These computations are akin to taking the derivative of a single-variable function in standard calculus. However, for multivariable functions, we perform this process for each variable in turn, which yields a set of partial derivatives—one for each variable, providing a broader picture of the function's behavior.
Partial Differentiation
Partial differentiation is the process of finding the partial derivatives of a multivariable function. This technique is analogous to differentiation of functions with one variable, but with a key difference: when differentiating with respect to a particular variable, all other variables are treated as constants.
Consider the given function
\(f(x, y, z)\).
The goal of partial differentiation is to determine how it responds to minor changes in 'x', 'y', or 'z' independently. Thus, for the function provided, partial differentiation yields three first partial derivatives, which were calculated in our example as follows:
Each expression encapsulates how the output of the function changes with an infinitesimal change in one of the inputs while the other inputs remain constant. Mastery of partial differentiation is crucial for students tackling complex problems in physics, engineering, economics, and beyond, as it lays the groundwork for optimizing functions, modelling natural phenomena, and understanding multivariable systems.
Consider the given function
\(f(x, y, z)\).
The goal of partial differentiation is to determine how it responds to minor changes in 'x', 'y', or 'z' independently. Thus, for the function provided, partial differentiation yields three first partial derivatives, which were calculated in our example as follows:
- \(\frac{\partial f}{\partial x} = yz + 2xz\)
- \(\frac{\partial f}{\partial y} = xz + 2xy + z^2\)
- \(\frac{\partial f}{\partial z} = xy + 2yz + x^2\)
Each expression encapsulates how the output of the function changes with an infinitesimal change in one of the inputs while the other inputs remain constant. Mastery of partial differentiation is crucial for students tackling complex problems in physics, engineering, economics, and beyond, as it lays the groundwork for optimizing functions, modelling natural phenomena, and understanding multivariable systems.
Other exercises in this chapter
Problem 20
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