How to Find the Derivative of (3x² + 5x - 2)/(x² + 1) Using the Quotient Rule
Master the quotient rule with this complete walkthrough of a real calculus problem. Learn the formula, avoid common mistakes, and practice with similar examples.
📹 Video Walkthrough: This Exact Problem
Watch the full solution for finding the derivative of (3x² + 5x - 2)/(x² + 1) step-by-step.
Table of Contents
The Problem
Find the derivative:
f(x) = (3x² + 5x - 2) / (x² + 1)
Use the quotient rule to find f'(x) and simplify your answer completely.
This is one of those classic calculus problems you'll see in homework and exams. You've got one polynomial sitting on top of another, and you need to find the derivative. That's exactly what the quotient rule was made for.
Sure, you could rewrite it as (3x² + 5x - 2) times (x² + 1)⁻¹ and use the product rule. But honestly? The quotient rule is way cleaner here. Let's stick with that. If you're stuck on a similar problem, you can always generate a custom video solution on Torial.
Understanding the Quotient Rule
Here's the quotient rule. If you've got a function that looks like f(x) = u(x) / v(x), the derivative is:
f'(x) = (v · u' - u · v') / v²
"low dee-high minus high dee-low, over low squared"
Most students remember it like this: "lo d-hi minus hi d-lo over lo squared."
- lo d-hi: bottom times derivative of top
- hi d-lo: top times derivative of bottom
- lo squared: bottom squared
💡 Important: Order matters here. It's low times the derivative of high MINUS high times the derivative of low. Not the other way around. Getting this backwards is the #1 mistake on exams. Don't be that person.
Identifying u and v
First step: figure out what's on top and what's on bottom. This seems obvious, but trust me, writing it down prevents mistakes later.
Given function:
f(x) = (3x² + 5x - 2) / (x² + 1)
Numerator (u):
u = 3x² + 5x - 2
Denominator (v):
v = x² + 1
Now we need u' and v'. Since both are just polynomials, we can use the power rule. Easy stuff. Need a refresher on basic derivatives? Check out other calculus videos in our library.
Finding u' and v'
Quick refresher: the power rule says the derivative of x^n is n·x^(n-1). Let's use that on each part.
Finding u' (derivative of the numerator):
u = 3x² + 5x - 2
Apply the power rule to each term:
- Derivative of 3x²: 2 · 3x¹ = 6x
- Derivative of 5x: 1 · 5x⁰ = 5
- Derivative of -2: 0 (constant)
u' = 6x + 5
Finding v' (derivative of the denominator):
v = x² + 1
Apply the power rule to each term:
- Derivative of x²: 2 · x¹ = 2x
- Derivative of 1: 0 (constant)
v' = 2x
Summary so far:
u = 3x² + 5x - 2
u' = 6x + 5
v = x² + 1
v' = 2x
Applying the Quotient Rule
Time to plug everything into the formula. This is where being organized actually matters.
The quotient rule formula:
f'(x) = (v · u' - u · v') / v²
Substituting our values:
f'(x) = [(x² + 1)(6x + 5) - (3x² + 5x - 2)(2x)] / (x² + 1)²
Breaking it down: v·u' is the first part, u·v' is the second, and v² goes on the bottom.
⚠️ Check before you expand: Make sure each piece is in the right spot. Swapping v·u' and u·v' is the fastest way to get the wrong answer.
Simplifying the Result
Now comes the algebra part. Expand everything, then combine like terms. Stay focused here.
Step 1: Expand (x² + 1)(6x + 5)
= x² · 6x + x² · 5 + 1 · 6x + 1 · 5
= 6x³ + 5x² + 6x + 5
Step 2: Expand (3x² + 5x - 2)(2x)
= 3x² · 2x + 5x · 2x + (-2) · 2x
= 6x³ + 10x² - 4x
Step 3: Subtract and combine like terms
Numerator = (6x³ + 5x² + 6x + 5) - (6x³ + 10x² - 4x)
Distribute the negative:
= 6x³ + 5x² + 6x + 5 - 6x³ - 10x² + 4x
Combine like terms:
- x³ terms: 6x³ - 6x³ = 0
- x² terms: 5x² - 10x² = -5x²
- x terms: 6x + 4x = 10x
- constants: 5
= -5x² + 10x + 5
Final Answer:
f'(x) = (-5x² + 10x + 5) / (x² + 1)²
Or, factoring out -5 from the numerator:
f'(x) = -5(x² - 2x - 1) / (x² + 1)²
See how the x³ terms canceled out? That happens a lot with quotient rule problems. Terms disappearing is totally normal. Just make sure your algebra is clean and you didn't drop any negatives. Want to see more worked examples? Browse through hundreds of math solutions on Torial.
Common Mistakes to Avoid
Here are the mistakes that cost students the most points. Learn them now so you don't make them on test day. If you want personalized help avoiding these errors, create a custom study video for your specific problem.
❌ Mistake #1: Flipping the Order
Writing (u·v' - v·u') instead of (v·u' - u·v'). This gives you the right numbers but with the wrong sign. Your answer will be exactly backwards.
Fix: Say "lo d-hi minus hi d-lo" out loud before you start. Write it down if you need to.
❌ Mistake #2: Forgetting the Minus Sign
Writing (v·u' + u·v') instead of (v·u' - u·v'). The quotient rule always has subtraction, never addition.
Fix: The formula is hardcoded with a minus. There's no situation where it becomes a plus.
❌ Mistake #3: Not Squaring the Denominator
Putting just v on the bottom instead of v². This breaks the whole formula.
Fix: It's "over lo squared." That means you square the entire bottom function, not just part of it.
❌ Mistake #4: Messing Up the Algebra
Dropping signs when you expand (3x² + 5x - 2)(2x). That -2 becomes -4x, not +4x. Easy to mess up when you're moving fast.
Fix: Write every single step. Don't skip lines trying to save time. You'll lose more points from mistakes than you'll save from speed.
❌ Mistake #5: Distributing the Negative Wrong
When you subtract (6x³ + 10x² - 4x), every sign flips. That -4x becomes +4x. Students forget this constantly.
Fix: Rewrite it as adding the opposite: +(-6x³) +(-10x²) +(+4x). Makes it impossible to mess up.
❌ Mistake #6: Trying to Cancel Terms
Trying to cancel x² from the top with x² from (x² + 1)² on the bottom. You can only cancel factors, not terms that are added or subtracted.
Fix: Only cancel if you can factor completely first. In this problem, nothing cancels. That's fine.
Practice Problems with Video Solutions
Best way to get good at this? Practice. Try these similar problems and check your work with the video solutions. You can also generate instant video explanations for any derivative problem you're working on.
Practice Problem 1: Similar Polynomials
Find the derivative: f(x) = (2x² + 3x + 1) / (x² + 4)
Hint: Same process. Find u', v', apply the formula. After you expand and combine, you should get 5x² + 16x + 12 in the numerator.
Get instant video solution on Torial →Practice Problem 2: Linear Denominator
Find the derivative: g(x) = (x³ - 2x) / (3x + 5)
Hint: Simpler bottom means simpler v' (just 3). Watch the algebra though.
Get instant video solution on Torial →Practice Problem 3: Higher Degree
Find the derivative: h(x) = (x⁴ + x²) / (x² - 1)
Hint: More terms to track, but same method. You might be able to factor and cancel after simplifying.
Get instant video solution on Torial →Practice Problem 4: With Trig Functions
Find the derivative: k(x) = (2x + 1) / sin(x)
Hint: Same quotient rule, but v' = cos(x) now. Remember: derivative of sin(x) is cos(x).
Get instant video solution on Torial →When to Use the Quotient Rule vs. Product Rule
Should you use the quotient rule or rewrite this as a product and use product rule instead?
✓ Use Quotient Rule When:
- It's already written as a fraction
- Top and bottom are both nice polynomials
- The problem says to use quotient rule
- You're comfortable with the formula
✓ Use Product Rule When:
- The bottom is super simple (like 1/x = x⁻¹)
- Product rule makes more sense to you
- It's easier to work with as a product
- You want extra practice with product rule
For this problem? Quotient rule wins. Rewriting as (3x² + 5x - 2)(x² + 1)⁻¹ means you'd need product rule AND chain rule. More work, more chances to mess up. When you're juggling multiple rules, it helps to have a step-by-step video walkthrough that shows exactly which rule to use when.
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