Find The Derivative Of $\log_6(x$\].A. $\frac{1}{6x}$ B. $\frac{1}{x}$ C. $\frac{1}{x \ln(6)}$ D. $\frac{\ln(6)}{x}$

Introduction

In this article, we will explore the concept of finding the derivative of a logarithmic function. Specifically, we will focus on finding the derivative of $\log_6(x)$. This is an important topic in calculus, as it has numerous applications in various fields such as physics, engineering, and economics.

What is a Logarithmic Function?

A logarithmic function is a function that is the inverse of an exponential function. In other words, if $y = a^x$, then $x = \log_a(y)$. The logarithmic function $\log_6(x)$ is a function that takes a positive real number $x$ as input and returns the exponent to which 6 must be raised to produce $x$.

The Derivative of a Logarithmic Function

To find the derivative of a logarithmic function, we can use the following formula:

$$\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}$$

where $a$ is the base of the logarithm.

Applying the Formula to $\log_6(x)$

Now, let's apply the formula to find the derivative of $\log_6(x)$.

$$\frac{d}{dx} \log_6(x) = \frac{1}{x \ln(6)}$$

Comparing the Answer to the Options

Now, let's compare our answer to the options provided.

• A. $\frac{1}{6x}$
• B. $\frac{1}{x}$
• C. $\frac{1}{x \ln(6)}$
• D. $\frac{\ln(6)}{x}$

Our answer, $\frac{1}{x \ln(6)}$, matches option C.

Conclusion

In this article, we have explored the concept of finding the derivative of a logarithmic function. We have applied the formula for the derivative of a logarithmic function to find the derivative of $\log_6(x)$, and we have compared our answer to the options provided. We have found that our answer matches option C.

Why is this Important?

Finding the derivative of a logarithmic function is an important topic in calculus because it has numerous applications in various fields such as physics, engineering, and economics. For example, in physics, the derivative of a logarithmic function can be used to model the behavior of a system that is subject to a logarithmic force. In engineering, the derivative of a logarithmic function can be used to design systems that are subject to logarithmic constraints.

Real-World Applications

The derivative of a logarithmic function has numerous real-world applications. For example:

• In physics, the derivative of a logarithmic function can be used to model the behavior of a system that is subject to a logarithmic force.
• In engineering, the derivative of a logarithmic function can be used to design systems that are subject to logarithmic constraints.
• In economics, the derivative of a logarithmic function can be used to model the behavior of a system that is subject to logarithmic constraints.

Common Mistakes to Avoid

When finding the derivative of a logarithmic function, there are several common mistakes to avoid. These include:

• Failing to apply the formula for the derivative of a logarithmic function.
• Failing to simplify the expression for the derivative.
• Failing to compare the answer to the options provided.

Tips and Tricks

When finding the derivative of a logarithmic function, there are several tips and tricks to keep in mind. These include:

• Make sure to apply the formula for the derivative of a logarithmic function.
• Make sure to simplify the expression for the derivative.
• Make sure to compare the answer to the options provided.

Conclusion

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Q: What is the derivative of $\log_6(x)$?

A: The derivative of $\log_6(x)$ is $\frac{1}{x \ln(6)}$.

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

A: To find the derivative of a logarithmic function, you can use the following formula:

$$\frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)}$$

where $a$ is the base of the logarithm.

Q: What is the significance of the base of the logarithm in the derivative formula?

A: The base of the logarithm is an important part of the derivative formula. It determines the rate at which the logarithmic function changes as the input variable changes.

Q: Can I use the derivative formula to find the derivative of any logarithmic function?

A: Yes, you can use the derivative formula to find the derivative of any logarithmic function. Simply substitute the base of the logarithm into the formula and simplify.

Q: How do I simplify the expression for the derivative?

A: To simplify the expression for the derivative, you can use the following steps:

1. Factor out any common terms.
2. Cancel out any common factors.
3. Simplify any remaining expressions.

Q: What are some common mistakes to avoid when finding the derivative of a logarithmic function?

A: Some common mistakes to avoid when finding the derivative of a logarithmic function include:

• Failing to apply the formula for the derivative of a logarithmic function.
• Failing to simplify the expression for the derivative.
• Failing to compare the answer to the options provided.

Q: Can I use the derivative of a logarithmic function to model real-world phenomena?

A: Yes, you can use the derivative of a logarithmic function to model real-world phenomena. For example, in physics, the derivative of a logarithmic function can be used to model the behavior of a system that is subject to a logarithmic force.

Q: What are some real-world applications of the derivative of a logarithmic function?

A: Some real-world applications of the derivative of a logarithmic function include:

• Modeling the behavior of a system that is subject to a logarithmic force.
• Designing systems that are subject to logarithmic constraints.
• Modeling the behavior of a system that is subject to logarithmic constraints.

Q: How do I apply the derivative of a logarithmic function to solve real-world problems?

A: To apply the derivative of a logarithmic function to solve real-world problems, you can use the following steps:

1. Identify the problem and the relevant variables.
2. Determine the type of logarithmic function that is relevant to the problem.
3. Apply the derivative formula to find the derivative of the logarithmic function.
4. Simplify the expression for the derivative.
5. Use the derivative to model the behavior of the system or to design a system that meets the constraints.

Q: What are some tips and tricks for finding the derivative of a logarithmic function?

A: Some tips and tricks for finding the derivative of a logarithmic function include:

• Make sure to apply the formula for the derivative of a logarithmic function.
• Make sure to simplify the expression for the derivative.
• Make sure to compare the answer to the options provided.
• Use the derivative formula to model real-world phenomena.
• Apply the derivative to solve real-world problems.