Derivative Of Ln(x^6 / (2x^5 - 7)) Step-by-Step Guide

Hey everyone! Today, we're diving into a fun calculus problem: finding the derivative of the function f(x) = ln(x⁶ / (2x⁵ - 7)). This might look a bit intimidating at first, but don't worry, we'll break it down step by step. We'll be using some key properties of logarithms and the chain rule to get to our final answer. So, grab your pencils, and let's get started!

Understanding the Function

Before we jump into the differentiation process, let's take a moment to understand what our function f(x) actually represents. We have a natural logarithm (ln) applied to a fraction. The numerator of the fraction is x⁶, which is simply x raised to the power of 6. The denominator is a bit more interesting: 2x⁵ - 7. This is a polynomial expression, where x is raised to the power of 5, multiplied by 2, and then we subtract 7. The entire fraction, x⁶ / (2x⁵ - 7), forms the argument of our natural logarithm.

So, essentially, f(x) is telling us how the natural logarithm changes as x varies, considering the ratio between x⁶ and 2x⁵ - 7. This kind of function can pop up in various contexts, like analyzing growth rates or modeling physical phenomena. The natural logarithm, in particular, is super useful because it has some nice properties that simplify calculations, as we'll see shortly.

The key here is to recognize that we're dealing with a composite function. We have an outer function (the natural logarithm) and an inner function (the fraction). This is a big hint that we'll need to use the chain rule when we differentiate. The chain rule, in a nutshell, tells us how to differentiate a composite function by breaking it down into its individual parts. We'll also leverage the properties of logarithms to make our lives easier. Logarithms have this cool ability to turn division into subtraction, which will simplify the expression inside the logarithm and make it easier to differentiate. We will see how using logarithm properties helps us solve this problem.

Step 1: Simplify Using Logarithmic Properties

The first thing we're going to do is simplify our function using the properties of logarithms. Remember, one of the key properties is that the logarithm of a quotient is equal to the difference of the logarithms. In other words, ln(a/b) = ln(a) - ln(b). This is going to be a game-changer for us!

Applying this property to our function, f(x) = ln(x⁶ / (2x⁵ - 7)), we get:

f(x) = ln(x⁶) - ln(2x⁵ - 7)

See how we've transformed a complex fraction inside the logarithm into a simple subtraction of two logarithms? That's the magic of logarithmic properties! Now, we can go even further. Another important property of logarithms states that ln(aᵇ) = b * ln(a). This means we can bring down the exponent in the first term.

Applying this property to ln(x⁶), we get 6ln(x). So our function now looks like this:

f(x) = 6ln(x) - ln(2x⁵ - 7)

Wow, that's a lot simpler, right? By using these two logarithmic properties, we've transformed our original function into a much more manageable form. We've essentially unwrapped the function, making it easier to see the individual components that we need to differentiate. This is a crucial step because it avoids dealing with the quotient rule directly inside the logarithm, which would have made things significantly more complicated. Now, we're ready to move on to the differentiation step. Remember, the power of logarithmic properties lies in their ability to simplify complex expressions, making them easier to handle in calculus.

Step 2: Differentiate Using the Chain Rule

Now that we've simplified our function, it's time to differentiate! We'll be using the chain rule, which is essential for differentiating composite functions like ours. Remember, the chain rule states that if we have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x). In simpler terms, we differentiate the outer function, keeping the inner function as it is, and then multiply by the derivative of the inner function.

Our function is now in the form f(x) = 6ln(x) - ln(2x⁵ - 7). Let's differentiate each term separately. For the first term, 6ln(x), we know that the derivative of ln(x) is 1/x. So, the derivative of 6ln(x) is simply 6 * (1/x) = 6/x.

Now, let's tackle the second term, -ln(2x⁵ - 7). This is where the chain rule comes into play. The outer function is ln(u), where u = 2x⁵ - 7, and the inner function is 2x⁵ - 7. The derivative of ln(u) with respect to u is 1/u. The derivative of 2x⁵ - 7 with respect to x is 10x⁴ (using the power rule). So, applying the chain rule, the derivative of -ln(2x⁵ - 7) is -(1/(2x⁵ - 7)) * (10x⁴) = -10x⁴ / (2x⁵ - 7).

Putting it all together, the derivative of our function f(x) is:

f'(x) = 6/x - 10x⁴ / (2x⁵ - 7)

And there you have it! We've successfully differentiated the function using the chain rule and our knowledge of logarithmic derivatives. The key here was to recognize the composite nature of the function and apply the chain rule carefully. Also, remember that the chain rule is a fundamental concept in calculus, and mastering it is crucial for differentiating a wide range of functions.

Step 3: Simplify (Optional)

While we've found the derivative, we can often simplify the expression further to make it cleaner and easier to work with. In this case, we have two terms with different denominators. To combine them, we can find a common denominator.

The common denominator for x and (2x⁵ - 7) is simply their product: x(2x⁵ - 7). So, let's rewrite each term with this common denominator:

6/x = (6(2x⁵ - 7)) / (x(2x⁵ - 7)) = (12x⁵ - 42) / (x(2x⁵ - 7))

-10x⁴ / (2x⁵ - 7) = (-10x⁴ * x) / (x(2x⁵ - 7)) = -10x⁵ / (x(2x⁵ - 7))

Now, we can combine the two terms:

f'(x) = (12x⁵ - 42 - 10x⁵) / (x(2x⁵ - 7))

Simplifying the numerator, we get:

f'(x) = (2x⁵ - 42) / (x(2x⁵ - 7))

This is a simplified form of our derivative. Whether you need to simplify further depends on the context of the problem. Sometimes, a simplified form is preferred for easier analysis or further calculations. Other times, the unsimplified form is perfectly acceptable. But remember simplifying the derivative can make it easier to analyze its behavior, such as finding critical points or intervals of increase and decrease.

The Final Answer

So, after simplifying the function using logarithmic properties, applying the chain rule, and combining terms with a common denominator, we've found the derivative of f(x) = ln(x⁶ / (2x⁵ - 7)). The derivative is:

f'(x) = (2x⁵ - 42) / (x(2x⁵ - 7))

Alternatively, the unsimplified form is:

f'(x) = 6/x - 10x⁴ / (2x⁵ - 7)

Both forms are correct, and the choice of which one to use often depends on the specific application. This problem highlights the importance of understanding logarithmic properties and the chain rule. These are powerful tools in calculus that allow us to tackle complex functions and find their derivatives. Remember, practice makes perfect! The more you work through these kinds of problems, the more comfortable you'll become with the techniques involved. Also, don't hesitate to use online resources, textbooks, or ask for help when you get stuck. Calculus can be challenging, but it's also incredibly rewarding when you master it. Keep up the great work, guys!

Key Takeaways

Before we wrap up, let's recap the key steps we took to find the derivative of our function:

1. Simplify Using Logarithmic Properties: We used the properties ln(a/b) = ln(a) - ln(b) and ln(aᵇ) = b * ln(a) to simplify the original function, making it easier to differentiate.
2. Differentiate Using the Chain Rule: We applied the chain rule to differentiate the composite function, breaking it down into its outer and inner components.
3. Simplify (Optional): We combined the terms with a common denominator to obtain a simplified form of the derivative.

By following these steps, we were able to successfully find the derivative of a seemingly complex function. Remember, the key to success in calculus is to break down problems into smaller, manageable steps. And don't forget to practice! The more you practice, the better you'll become at recognizing patterns and applying the appropriate techniques. Also, always remember the key takeaways and try to apply them in different scenarios. This will not only help you solve problems but also deepen your understanding of the concepts.

I hope this explanation was helpful! If you have any questions or want to explore other calculus problems, feel free to ask. Keep learning and keep exploring the fascinating world of mathematics!

Additional Tips for Differentiation

To further enhance your skills in differentiation, here are some additional tips that can prove beneficial:

• Master the Basic Derivatives: Ensure you have a strong grasp of the derivatives of basic functions such as polynomials, trigonometric functions, exponential functions, and logarithmic functions. This forms the foundation for tackling more complex problems.
• Practice, Practice, Practice: The more you practice differentiation problems, the more comfortable and proficient you will become. Work through a variety of examples to solidify your understanding.
• Use Online Resources: Numerous online resources, including websites, videos, and interactive tools, can aid in your learning process. Utilize these resources to supplement your understanding and explore different perspectives.
• Seek Help When Needed: Don't hesitate to ask for help from teachers, classmates, or online forums when you encounter difficulties. Collaboration and discussion can provide valuable insights and clarify concepts.

Remember, calculus is a journey of continuous learning and improvement. By following these tips and dedicating time and effort to practice, you can master differentiation and excel in your calculus studies. Good luck, guys!