Quotient Rule: Derivative Of F(x) = (8x + 9) / (4x + 7)

by TextBrain Team 56 views

Hey guys! In this article, we're going to dive deep into the quotient rule, a fundamental concept in calculus, and apply it to find the derivative of the function f(x) = (8x + 9) / (4x + 7). Don't worry if you're just starting with calculus; we'll break it down step-by-step so everyone can follow along. So, grab your pencils and let's get started!

Understanding the Quotient Rule

Before we jump into the specifics of our function, let's quickly recap what the quotient rule actually is. The quotient rule is a formula used to find the derivative of a function that is expressed as the quotient (or division) of two other functions. In simpler terms, if you have a function that looks like f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the quotient rule gives you a way to find f'(x), the derivative of f(x).

The formula for the quotient rule is as follows:

f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2

Where:

  • f'(x) is the derivative of the quotient function.
  • g(x) is the function in the numerator.
  • h(x) is the function in the denominator.
  • g'(x) is the derivative of g(x).
  • h'(x) is the derivative of h(x).

It might look a little intimidating at first, but trust me, it's quite straightforward once you get the hang of it. The key is to identify the g(x) and h(x) correctly and then carefully apply the formula. A handy way to remember it is: "(Low dHigh minus High dLow) over Low squared," where "Low" refers to the denominator and "High" refers to the numerator. This little mnemonic can save you some headaches when you're working through problems.

Understanding why the quotient rule works is just as crucial as knowing the formula itself. It stems from the product rule and the chain rule, which are other fundamental concepts in calculus. Essentially, the quotient rule is a shortcut derived from these more basic rules, making it more efficient to differentiate functions that are expressed as quotients. So, while memorizing the formula is important, always remember that it’s built on solid mathematical foundations. Understanding the underlying principles will help you apply it correctly in various situations and will deepen your overall understanding of calculus.

Applying the Quotient Rule to Our Function

Now, let's apply the quotient rule to our specific function, f(x) = (8x + 9) / (4x + 7). The first step is to identify our g(x) and h(x). In this case:

  • g(x) = 8x + 9 (the numerator)
  • h(x) = 4x + 7 (the denominator)

Next, we need to find the derivatives of g(x) and h(x). This is where the basic power rule of differentiation comes in handy. Remember, the power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). For constant multiples, like 8x, the derivative is just the constant, 8. The derivative of a constant, like 9 or 7, is zero.

So, let's find g'(x) and h'(x):

  • g'(x) = d/dx (8x + 9) = 8
  • h'(x) = d/dx (4x + 7) = 4

Great! We've got all the pieces we need. Now we can plug these into the quotient rule formula:

f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2

Substituting our values, we get:

f'(x) = [8(4x + 7) - (8x + 9)(4)] / (4x + 7)^2

Now, all that's left is to simplify the expression. This involves expanding the brackets and combining like terms. Be careful with your algebra here, guys; it's easy to make a small mistake that can throw off your entire answer. Always double-check your steps to make sure you're on the right track. Simplifying algebraic expressions is a fundamental skill in calculus, so practice makes perfect. The more comfortable you are with these manipulations, the easier it will be to tackle more complex calculus problems.

Simplifying the Derivative

Okay, let's simplify the derivative we found in the previous section. We have:

f'(x) = [8(4x + 7) - (8x + 9)(4)] / (4x + 7)^2

First, expand the terms in the numerator:

f'(x) = [32x + 56 - (32x + 36)] / (4x + 7)^2

Next, distribute the negative sign in the numerator:

f'(x) = [32x + 56 - 32x - 36] / (4x + 7)^2

Now, combine like terms in the numerator. Notice that the 32x and -32x terms cancel each other out:

f'(x) = [56 - 36] / (4x + 7)^2

Finally, subtract the constants in the numerator:

f'(x) = 20 / (4x + 7)^2

So, there you have it! The derivative of f(x) = (8x + 9) / (4x + 7), found using the quotient rule, is f'(x) = 20 / (4x + 7)^2. This simplified form is much cleaner and easier to work with for any further analysis or applications, such as finding critical points or analyzing the function's behavior. Remember, simplifying your answer is a crucial step in calculus problems. It not only gives you the most concise form of the solution but also reduces the chances of making errors in subsequent calculations. Plus, a simplified answer is often easier to interpret and understand conceptually.

Key Takeaways and Common Mistakes

Let's recap the key takeaways from this example and also touch on some common mistakes to avoid when using the quotient rule.

Key Takeaways:

  • The quotient rule is used to find the derivative of functions that are the quotient of two other functions.
  • The formula is: f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2.
  • Identify g(x) and h(x) correctly, then find their derivatives.
  • Carefully substitute into the formula and simplify.

Common Mistakes to Avoid:

  • Incorrectly identifying g(x) and h(x): Make sure you know which function is in the numerator (g(x)) and which is in the denominator (h(x)).
  • Forgetting the order in the numerator: The order of terms in the numerator matters! It's g'(x)h(x) - g(x)h'(x), not the other way around. Using the mnemonic "Low dHigh minus High dLow" can help with this.
  • Incorrectly finding derivatives: Make sure you're comfortable with basic differentiation rules like the power rule and the constant multiple rule.
  • Algebra errors in simplification: Be extra careful when expanding, distributing, and combining like terms. A small mistake here can lead to a completely wrong answer.
  • Forgetting to square the denominator: It's easy to overlook, but the denominator should be [h(x)]^2.

By keeping these key points and common mistakes in mind, you'll be well-equipped to tackle quotient rule problems with confidence. Practice is crucial, guys. The more you work through examples, the more natural the process will become.

Practice Problems

To solidify your understanding of the quotient rule, let's look at a couple of practice problems. Working through these will give you the hands-on experience you need to master this concept.

Practice Problem 1:

Find the derivative of f(x) = (3x^2 + 1) / (2x - 5)

Solution:

  1. Identify g(x) and h(x):
    • g(x) = 3x^2 + 1
    • h(x) = 2x - 5
  2. Find g'(x) and h'(x):
    • g'(x) = 6x
    • h'(x) = 2
  3. Apply the quotient rule formula:
    • f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2
    • f'(x) = [6x(2x - 5) - (3x^2 + 1)(2)] / (2x - 5)^2
  4. Simplify:
    • f'(x) = [12x^2 - 30x - 6x^2 - 2] / (2x - 5)^2
    • f'(x) = [6x^2 - 30x - 2] / (2x - 5)^2

Practice Problem 2:

Find the derivative of f(x) = (x) / (x^2 + 4)

Solution:

  1. Identify g(x) and h(x):
    • g(x) = x
    • h(x) = x^2 + 4
  2. Find g'(x) and h'(x):
    • g'(x) = 1
    • h'(x) = 2x
  3. Apply the quotient rule formula:
    • f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2
    • f'(x) = [1(x^2 + 4) - x(2x)] / (x^2 + 4)^2
  4. Simplify:
    • f'(x) = [x^2 + 4 - 2x^2] / (x^2 + 4)^2
    • f'(x) = [-x^2 + 4] / (x^2 + 4)^2

These practice problems should give you a better handle on applying the quotient rule. Remember to take your time, be careful with the algebra, and double-check your work. The key is to break the problem down into smaller steps and tackle each one methodically. Keep practicing, and you'll become a quotient rule pro in no time!

Conclusion

Alright, guys! We've covered a lot in this article. We started with a quick review of the quotient rule, discussed how to apply it to the function f(x) = (8x + 9) / (4x + 7), simplified the derivative, highlighted key takeaways and common mistakes, and even worked through a couple of practice problems. The quotient rule is a powerful tool in calculus, and mastering it will definitely boost your problem-solving abilities.

Remember, calculus, like any mathematical skill, requires consistent practice. Don't get discouraged if you find it challenging at first. Keep working at it, try different problems, and seek help when you need it. Understanding the underlying concepts and practicing regularly will make the quotient rule, and calculus in general, much more manageable.

So, keep practicing, stay curious, and happy calculating! You've got this!