Find The Derivative Of The Function.$\[ \begin{array}{l} g(x) = \ln (|7x - 4|) \\ g^{\prime}(x) = \square \end{array} \\]

Feb 27, 2025 by ADMIN 122 views

**Finding the Derivative of a Natural Logarithmic Function**

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a natural logarithmic function, specifically the function $g(x) = \ln (|7x - 4|)$ . This function involves the absolute value of a linear expression, which requires careful consideration when differentiating.

The Natural Logarithmic Function

The natural logarithmic function, denoted by $\ln x$ , is the inverse of the exponential function $e^x$ . It is defined for all positive real numbers and has a number of important properties, including:

$\ln 1 = 0$
$\ln e = 1$
$\ln (xy) = \ln x + \ln y$
$\ln (x/y) = \ln x - \ln y$

The Absolute Value Function

The absolute value function, denoted by $|x|$ , is defined as:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

This function is used to represent the distance of a number from zero on the number line.

Differentiating the Natural Logarithmic Function

To find the derivative of the function $g(x) = \ln (|7x - 4|)$ , we will use the chain rule and the fact that the derivative of the natural logarithmic function is $\frac{1}{x}$ .

Step 1: Differentiate the Inner Function

The inner function is $|7x - 4|$ . To differentiate this function, we will use the chain rule and the fact that the derivative of the absolute value function is $\frac{7}{|7x - 4|}$ .

\frac{d}{dx} |7x - 4| = \frac{7}{|7x - 4|}

Step 2: Apply the Chain Rule

The chain rule states that if we have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by:

\frac{d}{dx} f(g(x)) = f^{\prime}(g(x)) \cdot g^{\prime}(x)

In this case, we have:

g(x) = \ln (|7x - 4|)

f(x) = \ln x

f^{\prime}(x) = \frac{1}{x}

g^{\prime}(x) = \frac{7}{|7x - 4|}

Applying the chain rule, we get:

\frac{d}{dx} g(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)

\frac{d}{dx} g(x) = \frac{1}{\ln (|7x - 4|)} \cdot \frac{7}{|7x - 4|}

Simplifying the Derivative

We can simplify the derivative by canceling out the common factor of $|7x - 4|$ :

\frac{d}{dx} g(x) = \frac{7}{\ln (|7x - 4|) \cdot |7x - 4|}

\frac{d}{dx} g(x) = \frac{7}{(7x - 4) \ln (|7x - 4|)}

Conclusion

In this article, we have found the derivative of the natural logarithmic function $g(x) = \ln (|7x - 4|)$ . We used the chain rule and the fact that the derivative of the natural logarithmic function is $\frac{1}{x}$ to derive the derivative. The final derivative is $\frac{7}{(7x - 4) \ln (|7x - 4|)}$ .

Example Use Cases

The derivative of the natural logarithmic function has a number of important applications in mathematics and science. Some example use cases include:

Physics: The derivative of the natural logarithmic function is used to describe the rate of change of physical quantities such as temperature, pressure, and velocity.
Engineering: The derivative of the natural logarithmic function is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
Economics: The derivative of the natural logarithmic function is used to model and analyze economic systems, including the behavior of prices, wages, and interest rates.

References

Spivak, M. (2008). Calculus. Cambridge University Press.
Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
Guichard, D. (2013). Calculus: Single Variable. OpenStax.
Derivative of the Natural Logarithmic Function: Q&A =====================================================

Introduction

In our previous article, we discussed the derivative of the natural logarithmic function $g(x) = \ln (|7x - 4|)$ . In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the derivative of the natural logarithmic function?

A: The derivative of the natural logarithmic function $g(x) = \ln (|7x - 4|)$ is $\frac{7}{(7x - 4) \ln (|7x - 4|)}$ .

Q: How do I apply the chain rule to find the derivative of the natural logarithmic function?

A: To apply the chain rule, you need to identify the inner function and the outer function. In this case, the inner function is $|7x - 4|$ and the outer function is $\ln x$ . You then need to find the derivative of the inner function and the outer function separately, and multiply them together.

Q: What is the chain rule?

A: The chain rule is a fundamental concept in calculus that allows you to differentiate composite functions. It states that if you have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by:

\frac{d}{dx} f(g(x)) = f^{\prime}(g(x)) \cdot g^{\prime}(x)

Q: How do I simplify the derivative of the natural logarithmic function?

A: To simplify the derivative, you can cancel out any common factors. In this case, you can cancel out the common factor of $|7x - 4|$ :

\frac{d}{dx} g(x) = \frac{7}{\ln (|7x - 4|) \cdot |7x - 4|}

\frac{d}{dx} g(x) = \frac{7}{(7x - 4) \ln (|7x - 4|)}

Q: What are some example use cases of the derivative of the natural logarithmic function?

A: The derivative of the natural logarithmic function has a number of important applications in mathematics and science. Some example use cases include:

Physics: The derivative of the natural logarithmic function is used to describe the rate of change of physical quantities such as temperature, pressure, and velocity.
Engineering: The derivative of the natural logarithmic function is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
Economics: The derivative of the natural logarithmic function is used to model and analyze economic systems, including the behavior of prices, wages, and interest rates.

Q: Where can I find more information on the derivative of the natural logarithmic function?

A: There are many excellent resources available that provide detailed explanations and examples of the derivative of the natural logarithmic function. Some popular resources include:

Calculus Textbooks: There are many excellent calculus textbooks that cover the derivative of the natural logarithmic function in detail. Some popular textbooks include "Calculus" by Michael Spivak, "Calculus: Early Transcendentals" by James Stewart, and "Calculus: Single Variable" by David Guichard.
Online Resources: There are many online resources that provide detailed explanations and examples of the derivative of the natural logarithmic function. Some popular online resources include Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about the derivative of the natural logarithmic function. We hope that this article has been helpful in providing a better understanding of this important concept in calculus.

References

Spivak, M. (2008). Calculus. Cambridge University Press.
Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
Guichard, D. (2013). Calculus: Single Variable. OpenStax.

Find The Derivative Of The Function.$\[ \begin{array}{l} g(x) = \ln (|7x - 4|) \\ g^{\prime}(x) = \square \end{array} \\]

Introduction

The Natural Logarithmic Function

The Absolute Value Function

Differentiating the Natural Logarithmic Function

Step 1: Differentiate the Inner Function

Step 2: Apply the Chain Rule

Simplifying the Derivative

Conclusion

Example Use Cases

Further Reading

References

Introduction

Q: What is the derivative of the natural logarithmic function?

Q: How do I apply the chain rule to find the derivative of the natural logarithmic function?

Q: What is the chain rule?

Q: How do I simplify the derivative of the natural logarithmic function?

Q: What are some example use cases of the derivative of the natural logarithmic function?

Q: Where can I find more information on the derivative of the natural logarithmic function?

Conclusion

Further Reading

References