Hey there, math enthusiasts! Ever find yourself staring at a function and wondering how to find its derivative? You're not alone! Derivatives might seem intimidating at first, but with the right tools and a little practice, you'll be a pro in no time. In this article, we're going to break down a classic calculus problem step by step, so you can confidently tackle similar problems in the future. We will explore in detail how to find the derivative, f'(x), of the function f(x) = (x^2 + 4)/(9x - 1) and simplify it. So, grab your pencils and let’s dive in!
Understanding the Problem
Before we jump into the solution, let's make sure we understand what the problem is asking. We're given a function, f(x) = (x^2 + 4)/(9x - 1), which is a rational function (a fraction where both the numerator and denominator are polynomials). Our mission, should we choose to accept it, is to find the derivative of this function, denoted as f'(x). The derivative represents the instantaneous rate of change of the function at any given point. In simpler terms, it tells us how the function's output changes as its input changes. To find the derivative of this particular function, we'll need to employ a powerful tool called the Quotient Rule. This rule is specifically designed for finding the derivatives of functions that are expressed as the quotient (or fraction) of two other functions. Let's explore how this rule works and why it's essential for solving our problem.
Why the Quotient Rule?
You might be wondering, “Why can't we just take the derivative of the numerator and denominator separately?” Great question! The derivative of a quotient isn't as simple as that. The Quotient Rule takes into account how both the numerator and the denominator contribute to the overall rate of change of the function. It ensures we're capturing the complete picture of the derivative. Think of it like this: imagine you're running a race. Your speed depends not only on how fast you're moving forward but also on whether you're running uphill or downhill. The Quotient Rule is like the calculus equivalent of considering both your forward speed and the slope of the terrain.
In mathematical terms, if we have a function f(x) that can be written as f(x) = u(x) / v(x), where u(x) and v(x) are differentiable functions (meaning they have derivatives), then the Quotient Rule states:
f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
This formula might look a bit intimidating at first glance, but don't worry! We'll break it down step by step and apply it to our function. The key is to correctly identify u(x) and v(x) in our given function and then find their respective derivatives, u'(x) and v'(x). Once we have these pieces, we can plug them into the formula and simplify to find f'(x). Now, let’s move on to identifying the parts of our function and applying the Quotient Rule.
Applying the Quotient Rule
Okay, let's get our hands dirty and apply the Quotient Rule to our function, f(x) = (x^2 + 4)/(9x - 1). The first step, as we discussed, is to correctly identify u(x) and v(x). Remember, u(x) is the numerator (the top part of the fraction), and v(x) is the denominator (the bottom part of the fraction). In our case:
- u(x) = x^2 + 4
- v(x) = 9x - 1
Now that we've identified u(x) and v(x), the next crucial step is to find their derivatives, u'(x) and v'(x). To do this, we'll use the power rule and the constant rule of differentiation. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1)*. The constant rule states that the derivative of a constant is zero. Let's apply these rules to find u'(x) and v'(x).
Finding u'(x)
u(x) = x^2 + 4
To find u'(x), we'll differentiate each term in u(x) separately. The derivative of x^2 is 2x (using the power rule), and the derivative of the constant 4 is 0 (using the constant rule). Therefore:
u'(x) = 2x + 0 = 2x
Finding v'(x)
v(x) = 9x - 1
Similarly, to find v'(x), we'll differentiate each term in v(x). The derivative of 9x is 9 (since the derivative of x is 1), and the derivative of the constant -1 is 0. So:
v'(x) = 9 - 0 = 9
Great! We've successfully found u(x), v(x), u'(x), and v'(x). Now we have all the ingredients we need to plug into the Quotient Rule formula. Remember the formula?
f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
Let's substitute our values into this formula and see what we get.
Plugging into the Quotient Rule Formula
Now comes the satisfying part – putting all our hard work together! We'll substitute the expressions we found for u(x), v(x), u'(x), and v'(x) into the Quotient Rule formula:
f'(x) = [(9x - 1) * (2x) - (x^2 + 4) * (9)] / (9x - 1)^2
Take a moment to appreciate this step. We've successfully applied the Quotient Rule and expressed f'(x) in terms of our original functions and their derivatives. However, we're not quite done yet. The next step is crucial: we need to simplify this expression. Simplification is not just about making the expression look neater; it often reveals hidden structures and makes it easier to work with the derivative in later calculations, such as finding critical points or analyzing the function's behavior. So, let's roll up our sleeves and dive into the simplification process.
Simplifying the Expression
The expression we obtained after applying the Quotient Rule looks a bit messy, right? Don't worry; simplification is our friend! It's like tidying up a room – once we've organized things, we can see the beauty and order within. To simplify our expression, f'(x) = [(9x - 1) * (2x) - (x^2 + 4) * (9)] / (9x - 1)^2, we'll need to perform a few algebraic manipulations. The key techniques we'll use are expanding the products in the numerator and then combining like terms. Let's break it down step by step.
Expanding the Products
The first step in simplifying the numerator is to expand the products. This means multiplying out the terms within the parentheses. Let's start with the first product, (9x - 1) * (2x). We'll use the distributive property (which, in simpler terms, means multiplying each term inside the first set of parentheses by each term inside the second set):
(9x - 1) * (2x) = (9x * 2x) + (-1 * 2x) = 18x^2 - 2x
Now let's expand the second product, (x^2 + 4) * (9). Again, we'll use the distributive property:
(x^2 + 4) * (9) = (x^2 * 9) + (4 * 9) = 9x^2 + 36
So, after expanding the products, our numerator looks like this:
18x^2 - 2x - (9x^2 + 36)
Notice the minus sign in front of the second expression. This is super important! We need to distribute this minus sign to both terms inside the parentheses. This is a common place where mistakes happen, so let's be extra careful. Distributing the minus sign gives us:
18x^2 - 2x - 9x^2 - 36
Now that we've expanded the products and distributed the minus sign, we're ready for the next step: combining like terms. This will help us to further simplify the numerator and get it into its most compact form.
Combining Like Terms
Combining like terms is like sorting socks in your laundry – you group together the ones that are similar. In our expression, 18x^2 - 2x - 9x^2 - 36, the “like terms” are the ones with the same variable raised to the same power. So, we can combine the x^2 terms (18x^2 and -9x^2) and the constant terms (-36). The -2x term is unique, so it will stay as it is.
Let's combine the x^2 terms:
18x^2 - 9x^2 = 9x^2
Now, let's put everything together. Our simplified numerator becomes:
9x^2 - 2x - 36
Wow, that looks much better! We've successfully simplified the numerator by expanding the products and combining like terms. But remember, the denominator is still there, waiting for us. Let's not forget about it! Our expression for f'(x) now looks like this:
f'(x) = (9x^2 - 2x - 36) / (9x - 1)^2
The denominator is (9x - 1)^2. While we could expand this, it's often best to leave it in factored form unless there's a specific reason to expand it. Leaving it factored can be helpful for things like finding critical points or analyzing the function's asymptotes later on. So, for now, we'll leave the denominator as it is. We have now arrived at the simplified form of f'(x). But before we declare victory, let's take a final look and make sure there's nothing else we can simplify. In this case, the numerator and denominator don't share any common factors, so we're indeed done!
The Final Answer
After all our hard work, we've arrived at the final answer! The derivative of f(x) = (x^2 + 4)/(9x - 1) is:
f'(x) = (9x^2 - 2x - 36) / (9x - 1)^2
Congratulations! You've successfully navigated the Quotient Rule and simplified the expression to find the derivative. This is a significant accomplishment in calculus, and you should be proud of yourself. Remember, practice makes perfect. The more you work with derivatives, the more comfortable and confident you'll become. Now, let's briefly discuss a common mistake that students often make when applying the Quotient Rule and how to avoid it.
Avoiding Common Mistakes
Calculus can be tricky, and even the most seasoned mathematicians make mistakes from time to time. One common pitfall when applying the Quotient Rule is getting the order of the terms in the numerator wrong. Remember the Quotient Rule formula:
f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
The minus sign in the numerator is the key here. It dictates the order of the terms. It's crucial to subtract u(x) * v'(x) from v(x) * u'(x), not the other way around. If you reverse the order, you'll end up with the wrong sign for your derivative, which can lead to incorrect conclusions later on. So, always double-check that you're applying the formula in the correct order.
Another common mistake is forgetting to distribute the minus sign correctly when simplifying the expression. As we saw earlier, when we have an expression like a - (b + c), we need to distribute the minus sign to both b and c, resulting in a - b - c. Failing to do this can lead to incorrect simplification and an incorrect final answer. To avoid this mistake, take your time, write out each step clearly, and double-check your work.
Finally, don't forget the importance of simplification! While it might be tempting to stop after applying the Quotient Rule, simplifying the expression is often necessary to get the derivative into its most usable form. A simplified derivative is easier to analyze, use for further calculations, and compare with other results. So, always make simplification a part of your problem-solving process.
Conclusion
Finding the derivative of a function using the Quotient Rule might seem challenging at first, but as we've seen, it's a manageable process when broken down into steps. We started by understanding the problem and recognizing the need for the Quotient Rule. Then, we carefully identified u(x) and v(x), found their derivatives, plugged everything into the Quotient Rule formula, and meticulously simplified the expression. The result is the derivative of our function, f'(x) = (9x^2 - 2x - 36) / (9x - 1)^2. Remember, the key to mastering calculus is practice. Work through plenty of examples, pay attention to the details, and don't be afraid to ask for help when you need it. With dedication and perseverance, you'll conquer even the most challenging calculus problems. Keep practicing, keep exploring, and keep learning! You've got this!