The Properties of the Logarithm

Oct 13, 2025 by windailys.com 129 views

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. The logarithmic function is a fundamental function in mathematics, and its derivative is a crucial concept in calculus. In this article, we will explore the derivative of the function $f(x)=\ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)$ using the properties of the logarithm.

The Properties of the Logarithm

Before we dive into the derivative of the given function, let's recall the properties of the logarithm. The logarithmic function is defined as:

\log_b(x) = \frac{\log_a(x)}{\log_a(b)}

where $a$ and $b$ are positive real numbers, and $x$ is a positive real number. The logarithmic function has several properties, including:

Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
Power Rule: $\log_b(x^n) = n\log_b(x)$

Derivative of the Logarithmic Function

To find the derivative of the given function, we will use the chain rule and the product 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'(g(x)) \cdot g'(x)

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

\frac{d}{dx}f(x)g(x) = f'(x)g(x) + f(x)g'(x)

Derivative of the Given Function

Now, let's find the derivative of the given function using the chain rule and the product rule.

f(x) = \ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)

To find the derivative of the function, we will first find the derivative of the argument of the logarithmic function, which is:

\frac{d}{dx}\left(\frac{7 x \sin (4 x)}{3 x+5}\right)

Using the quotient rule, we get:

\frac{d}{dx}\left(\frac{7 x \sin (4 x)}{3 x+5}\right) = \frac{(3 x+5)(7 \sin (4 x) + 28 x \cos (4 x)) - (7 x \sin (4 x))(3)}{(3 x+5)^2}

Simplifying the expression, we get:

\frac{d}{dx}\left(\frac{7 x \sin (4 x)}{3 x+5}\right) = \frac{(3 x+5)(7 \sin (4 x) + 28 x \cos (4 x)) - 21 x \sin (4 x)}{(3 x+5)^2}

Now, we can find the derivative of the logarithmic function using the chain rule:

f'(x) = \frac{1}{\frac{7 x \sin (4 x)}{3 x+5}} \cdot \frac{d}{dx}\left(\frac{7 x \sin (4 x)}{3 x+5}\right)

Substituting the expression for the derivative of the argument of the logarithmic function, we get:

f'(x) = \frac{1}{\frac{7 x \sin (4 x)}{3 x+5}} \cdot \frac{(3 x+5)(7 \sin (4 x) + 28 x \cos (4 x)) - 21 x \sin (4 x)}{(3 x+5)^2}

Simplifying the expression, we get:

f'(x) = \frac{(3 x+5)(7 \sin (4 x) + 28 x \cos (4 x)) - 21 x \sin (4 x)}{(3 x+5)^2 \cdot \frac{7 x \sin (4 x)}{3 x+5}}

Simplifying the Expression

To simplify the expression, we can cancel out the common factors in the numerator and denominator:

f'(x) = \frac{(3 x+5)(7 \sin (4 x) + 28 x \cos (4 x)) - 21 x \sin (4 x)}{7 x \sin (4 x) (3 x+5)}

Simplifying the expression further, we get:

f'(x) = \frac{21 \sin (4 x) + 84 x \cos (4 x) - 21 x \sin (4 x)}{7 x \sin (4 x) (3 x+5)}

Combining like terms, we get:

f'(x) = \frac{21 \sin (4 x) - 21 x \sin (4 x) + 84 x \cos (4 x)}{7 x \sin (4 x) (3 x+5)}

Factoring out the common term $21 \sin (4 x)$ , we get:

f'(x) = \frac{21 \sin (4 x) (1 - x) + 84 x \cos (4 x)}{7 x \sin (4 x) (3 x+5)}

Final Answer

The final answer is:

f'(x) = \frac{21 \sin (4 x) (1 - x) + 84 x \cos (4 x)}{7 x \sin (4 x) (3 x+5)}

This is the derivative of the given function using the properties of the logarithm.

Conclusion

In this article, we have found the derivative of the function $f(x)=\ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)$ using the properties of the logarithm. We have used the chain rule and the product rule to find the derivative of the function. The final answer is a complex expression involving trigonometric functions and rational functions. This derivative is a crucial concept in calculus and has many applications in physics, engineering, and economics.

Introduction

In our previous article, we explored the derivative of the function $f(x)=\ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)$ using the properties of the logarithm. In this article, we will provide a Q&A guide to help you understand the concept of the derivative of a logarithmic function.

Q: What is the derivative of a logarithmic function?

A: The derivative of a logarithmic function is a measure of how fast the function changes as its input changes. In other words, it represents the rate of change of the function with respect to one of its variables.

Q: How do you find the derivative of a logarithmic function?

A: To find the derivative of a logarithmic function, you can use the chain rule and the product rule. The chain rule 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'(g(x)) \cdot g'(x)

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

\frac{d}{dx}f(x)g(x) = f'(x)g(x) + f(x)g'(x)

Q: What is the chain rule?

A: The chain rule is a fundamental concept in calculus that allows you to find the derivative of a composite function. 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'(g(x)) \cdot g'(x)

Q: What is the product rule?

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

\frac{d}{dx}f(x)g(x) = f'(x)g(x) + f(x)g'(x)

Q: How do you apply the chain rule and the product rule to find the derivative of a logarithmic function?

A: To apply the chain rule and the product rule to find the derivative of a logarithmic function, you need to follow these steps:

Identify the argument of the logarithmic function.
Find the derivative of the argument using the chain rule and the product rule.
Use the chain rule to find the derivative of the logarithmic function.

Q: What is the derivative of the function $f(x)=\ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)$ ?

A: The derivative of the function $f(x)=\ln \left(\frac{7 x \sin (4 x)}{3 x+5}\right)$ is:

f'(x) = \frac{21 \sin (4 x) (1 - x) + 84 x \cos (4 x)}{7 x \sin (4 x) (3 x+5)}

Q: What are some common applications of the derivative of a logarithmic function?

A: The derivative of a logarithmic function has many applications in physics, engineering, and economics. Some common applications include:

Modeling population growth and decay
Modeling chemical reactions
Modeling electrical circuits
Modeling financial markets

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of the derivative of a logarithmic function. We have covered topics such as the chain rule, the product rule, and the derivative of a logarithmic function. We hope that this guide has been helpful in understanding the concept of the derivative of a logarithmic function.

Additional Resources

If you are interested in learning more about the derivative of a logarithmic function, we recommend the following resources:

Calculus textbooks
Online calculus courses
Calculus tutorials on YouTube
Calculus forums and communities

We hope that this guide has been helpful in understanding the concept of the derivative of a logarithmic function. If you have any further questions or need additional help, please don't hesitate to ask.