Error Spotting: Derivative Calculation Of (2x+5)/(x^2-1)

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Hey guys! Let's dive into a common calculus problem and sharpen our error-detecting skills. We're going to analyze a worked-out derivative calculation and pinpoint exactly where the mistake was made. This is super important because understanding why an error occurred is just as valuable as knowing the correct solution. It helps us build a stronger foundation and avoid similar slips in the future. So, grab your thinking caps, and let's get started!

The Problematic Derivative

Here's the problem we're tackling. Our mission is to find the derivative of the function (2x+5)/(x^2-1). The provided solution attempts to use the quotient rule, which is the correct approach for this type of problem. Let's take a look at the steps:

D_x\left(\frac{2 x+5}{x^2-1}\right) =\frac{(2 x+5)(2 x)-\left(x^2-1\right) 2}{\left(x^2-1\right)^2}

= \frac{4 x^2+10 x-2 x^2+2}{\left(x^2-1\right)^2}

= \frac{2 x^2+10}{\left(x^2-1\right)^2}

At first glance, it might seem okay, but a closer examination reveals a critical error. Let's break down the quotient rule first to understand what should have happened.

Understanding the Quotient Rule

The quotient rule is your best friend when you need to find the derivative of a function that's expressed as a fraction – one function divided by another. It states that if you have a function f(x) = u(x) / v(x), then its derivative f'(x) is given by:

f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2

Where:

  • u(x) is the function in the numerator
  • v(x) is the function in the denominator
  • u'(x) is the derivative of u(x)
  • v'(x) is the derivative of v(x)

This formula might look a bit intimidating at first, but it’s actually quite straightforward once you get the hang of it. The key is to correctly identify u(x), v(x), and their respective derivatives. Let's apply this to our problem and see where things went wrong in the provided solution. In our case:

  • u(x) = 2x + 5
  • v(x) = x^2 - 1

Now, we need to find the derivatives of these functions:

  • u'(x) = 2 (The derivative of 2x + 5 is simply 2)
  • v'(x) = 2x (The derivative of x^2 - 1 is 2x)

Okay, we've got all the pieces. Now, let's carefully plug them into the quotient rule formula and compare our result with the provided solution. This is where we'll be able to pinpoint the exact mistake.

Spotting the Flaw: Applying the Quotient Rule Correctly

Now let's correctly apply the quotient rule to our function f(x) = (2x + 5) / (x^2 - 1). We've already identified:

  • u(x) = 2x + 5
  • v(x) = x^2 - 1
  • u'(x) = 2
  • v'(x) = 2x

Plugging these into the quotient rule formula, f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2, we get:

f'(x) = [(x^2 - 1) * 2 - (2x + 5) * 2x] / (x^2 - 1)^2

Now, let's carefully expand and simplify the numerator:

f'(x) = [2x^2 - 2 - (4x^2 + 10x)] / (x^2 - 1)^2

f'(x) = [2x^2 - 2 - 4x^2 - 10x] / (x^2 - 1)^2

f'(x) = [-2x^2 - 10x - 2] / (x^2 - 1)^2

Now, let's compare this correct result with the initial attempt. Looking back at the first step of the incorrect solution:

D_x\left(\frac{2 x+5}{x^2-1}\right) =\frac{(2 x+5)(2 x)-\left(x^2-1\right) 2}{\left(x^2-1\right)^2}

The error lies in the incorrect application of the quotient rule's numerator. The terms (2x+5) and (2x) are multiplied in the wrong order. It looks like the solution tried to calculate u(x) * v'(x) in the first term of the numerator, but placed it in the position where v(x) * u'(x) should be. This seemingly small mistake throws off the entire calculation.

The Correct Solution Step-by-Step

Let's reiterate the correct steps to solve this derivative problem. This will solidify our understanding and highlight the importance of meticulousness in calculus.

  1. Identify u(x) and v(x):
    • u(x) = 2x + 5
    • v(x) = x^2 - 1
  2. Find u'(x) and v'(x):
    • u'(x) = 2
    • v'(x) = 2x
  3. Apply the Quotient Rule:
    • f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2
    • f'(x) = [(x^2 - 1) * 2 - (2x + 5) * 2x] / (x^2 - 1)^2
  4. Expand and Simplify:
    • f'(x) = [2x^2 - 2 - 4x^2 - 10x] / (x^2 - 1)^2
    • f'(x) = [-2x^2 - 10x - 2] / (x^2 - 1)^2

So, the correct derivative of (2x + 5) / (x^2 - 1) is (-2x^2 - 10x - 2) / (x^2 - 1)^2. We can even simplify this further by factoring out a -2 from the numerator:

f'(x) = -2(x^2 + 5x + 1) / (x^2 - 1)^2

Why is Error Analysis Important?

Guys, finding errors isn't just about correcting a single problem. It's a crucial skill in mathematics and beyond! Here’s why:

  • Deeper Understanding: When you analyze an error, you're forced to revisit the underlying concepts and principles. This leads to a more thorough and lasting understanding.
  • Preventing Future Mistakes: Identifying why an error occurred helps you avoid similar mistakes in the future. It's about learning from your slip-ups and solidifying your knowledge.
  • Critical Thinking: Error analysis sharpens your critical thinking skills. You learn to question assumptions, examine steps meticulously, and develop a more analytical approach to problem-solving.
  • Problem-Solving Prowess: The ability to troubleshoot and identify errors is a valuable asset in any field. It demonstrates a strong grasp of the subject matter and the ability to think logically.

Key Takeaways

Let's wrap up what we've learned in this exploration of derivative error spotting:

  • The Quotient Rule: Remember the formula: f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. Make sure you identify u(x), v(x), and their derivatives correctly.
  • Order Matters: In the quotient rule, the order of terms in the numerator is crucial. Getting it wrong, as we saw, leads to an incorrect result.
  • Simplify Carefully: After applying the quotient rule, take your time to expand and simplify the expression correctly. Watch out for those pesky negative signs!
  • Error Analysis is a Superpower: Make error analysis a regular part of your learning process. It's one of the most effective ways to strengthen your understanding and prevent future mistakes.

So, next time you're working on a math problem, don't just focus on getting the right answer. Also, think about why you might get something wrong. Analyze potential errors and develop a habit of double-checking your work. This will transform you from a simple problem-solver into a true mathematical master! Keep practicing, keep questioning, and keep learning! You guys got this!