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Magazine Chapter 5: Problem 16
A function of three variables is given by $$ f\left(x_{1}, x_{2}, x_{3}\right)=\frac{x_{1} x_{3}^{2}}{x_{2}}+\ln \left(x_{2} x_{3}\right) $$ Find all of the first- and second-order derivatives of this function and verify that \(f_{12}=f_{21}, \quad f_{13}=f_{31} \quad\) and \(f_{23}=f_{32}\)
Short Answer
Expert verified
The first-order partial derivatives are: \(\frac{x_3^2}{x_2}\), \(-\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2}\), \(\frac{2 x_1 x_3}{x_2} + \frac{1}{x_3}\). The second-order derivatives verify the symmetry: \(f_{12} = f_{21}\), \(f_{13} = f_{31}\), \(f_{23} = f_{32}\).
Step by step solution
01
Find the first-order partial derivatives
First, find the partial derivative of the function with respect to each variable: \[ f(x_1, x_2, x_3) = \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \]
02
Step 1.1: Partial derivative with respect to x1
For \( x_1 \): \[ \frac{\partial f}{\partial x_1} = \frac{\partial}{\partial x_1} \left( \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \right) = \frac{x_3^2}{x_2} \]
03
Step 1.2: Partial derivative with respect to x2
For \( x_2 \): \[ \frac{\partial f}{\partial x_2} = \frac{\partial}{\partial x_2} \left( \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \right) = -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \]
04
Step 1.3: Partial derivative with respect to x3
For \( x_3 \): \[ \frac{\partial f}{\partial x_3} = \frac{\partial}{\partial x_3} \left( \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \right) = \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \]
05
Find the second-order partial derivatives
Next, calculate the second-order partial derivatives of the function.
06
Step 2.1: Second-order partial derivative \(f_{11}\)
Since \( f_{x_1} = \frac{x_3^2}{x_2} \) does not depend on \( x_1 \): \[ \frac{\partial^2 f}{\partial x_1^2} = 0 \]
07
Step 2.2: Second-order partial derivative \(f_{12}\)
Calculate \( f_{12} \): \[ \frac{\partial}{\partial x_2} \left( \frac{x_3^2}{x_2} \right) = -\frac{x_3^2}{x_2^2} \]
08
Step 2.3: Second-order partial derivative \(f_{13}\)
Calculate \( f_{13} \): \[ \frac{\partial}{\partial x_3} \left( \frac{x_3^2}{x_2} \right) = \frac{2 x_3}{x_2} \]
09
Step 2.4: Second-order partial derivative \(f_{22}\)
Using \( f_{x_2} = -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \): \[ \frac{\partial^2 f}{\partial x_2^2} = \frac{2 x_1 x_3^2}{x_2^3} - \frac{1}{x_2^2} \]
10
Step 2.5: Second-order partial derivative \(f_{23}\)
Calculate \( f_{23} \): \[ \frac{\partial}{\partial x_3} \left( -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \right) = -\frac{2 x_1 x_3}{x_2^2} \]
11
Step 2.6: Second-order partial derivative \(f_{33}\)
Using \( f_{x_3} = \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \): \[ \frac{\partial^2 f}{\partial x_3^2} = \frac{2 x_1}{x_2} - \frac{1}{x_3^2} \]
12
Verify mixed partial derivatives
Verify that the mixed partial derivatives are equal.
13
Step 3.1: Verify \(f_{12} = f_{21}\)
Calculate \( f_{21} \): \[ \frac{\partial}{\partial x_1} \left( -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \right) = -\frac{x_3^2}{x_2^2} \] Since \( f_{12} = -\frac{x_3^2}{x_2^2} \), we verify that \( f_{12} = f_{21} \).
14
Step 3.2: Verify \(f_{13} = f_{31}\)
Calculate \( f_{31} \): \[ \frac{\partial}{\partial x_1} \left( \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \right) = \frac{2 x_3}{x_2} \] Since \( f_{13} = \frac{2 x_3}{x_2} \), we verify that \( f_{13} = f_{31} \).
15
Step 3.3: Verify \(f_{23} = f_{32}\)
Calculate \( f_{32} \): \[ \frac{\partial}{\partial x_2} \left( \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \right) = -\frac{2 x_1 x_3}{x_2^2} \] Since \( f_{23} = -\frac{2 x_1 x_3}{x_2^2} \), we verify that \( f_{23} = f_{32} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First-Order Derivatives
The first-order derivatives, also known as partial derivatives, measure how a function changes as its variables change. For a function of three variables like \(f(x_1, x_2, x_3)\), you would calculate the partial derivative with respect to each variable while holding the others constant.
In our example, we start with the given function: \[ f(x_1, x_2, x_3) = \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \] The first-order partial derivatives are:
In our example, we start with the given function: \[ f(x_1, x_2, x_3) = \frac{x_1 x_3^2}{x_2} + \ln(x_2 x_3) \] The first-order partial derivatives are:
- With respect to \(x_1\): \[ \frac{\partial f}{\partial x_1} = \frac{x_3^2}{x_2} \]
- With respect to \(x_2\): \[ \frac{\partial f}{\partial x_2} = -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \]
- With respect to \(x_3\): \[ \frac{\partial f}{\partial x_3} = \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \]
Second-Order Derivatives
Second-order derivatives are the partial derivatives of the first-order partial derivatives. They provide deeper insight into the curvature and concavity/convexity of a function.
For our function \(f(x_1, x_2, x_3)\), the second-order derivatives include:
For our function \(f(x_1, x_2, x_3)\), the second-order derivatives include:
- Second partial derivative with respect to \(x_1\): \[ \frac{\partial^2 f}{\partial x_1^2} = 0 \] Since the first partial derivative \( \frac{\partial f}{\partial x_1} = \frac{x_3^2}{x_2} \) does not depend on \(x_1\).
- Second partial derivative with respect to \(x_2\): \[ \frac{\partial^2 f}{\partial x_2^2} = \frac{2 x_1 x_3^2}{x_2^3} - \frac{1}{x_2^2} \]
- Second partial derivative with respect to \(x_3\): \[ \frac{\partial^2 f}{\partial x_3^2} = \frac{2 x_1}{x_2} - \frac{1}{x_3^2} \]
- \( f_{12} = \frac{\partial}{\partial x_2} \left( \frac{x_3^2}{x_2} \right) = -\frac{x_3^2}{x_2^2} \)
- \( f_{13} = \frac{\partial}{\partial x_3} \left( \frac{x_3^2}{x_2} \right) = \frac{2 x_3}{x_2} \)
- \( f_{23} = \frac{\partial}{\partial x_3} \left( -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \right) = -\frac{2 x_1 x_3}{x_2^2} \)
Mixed Partial Derivatives
Mixed partial derivatives refer to partial derivatives taken with respect to more than one variable, in different orders. In other words, you take the partial derivative of a function with respect to one variable, and then take the partial derivative of that result with respect to another variable. A key property of mixed partial derivatives, under certain regularity conditions, is that they are equal regardless of the order in which you differentiate.
For our function \(f(x_1, x_2, x_3)\), we verify the equality of mixed partial derivatives:
For our function \(f(x_1, x_2, x_3)\), we verify the equality of mixed partial derivatives:
- Verify \(f_{12} = f_{21}\): \[ \frac{\partial}{\partial x_1} \left( -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \right) = -\frac{x_3^2}{x_2^2} \] This matches \( \frac{\partial}{\partial x_2} \left( \frac{x_3^2}{x_2} \right)\).
- Verify \(f_{13} = f_{31}\): \[ \frac{\partial}{\partial x_1} \left( \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \right) = \frac{2 x_3}{x_2} \] This matches \( \frac{\partial}{\partial x_3} \left( \frac{x_3^2}{x_2} \right)\).
- Verify \(f_{23} = f_{32}\): \[ \frac{\partial}{\partial x_2} \left( \frac{2 x_1 x_3}{x_2} + \frac{1}{x_3} \right) = -\frac{2 x_1 x_3}{x_2^2} \] This matches \( \frac{\partial}{\partial x_3} \left( -\frac{x_1 x_3^2}{x_2^2} + \frac{1}{x_2} \right)\).
