Introduction
In O-Level A-Maths, quotient rule questions often test whether students can set up u and v correctly, differentiate carefully, and simplify neatly.
In this question, we differentiate a function in the form \(y=\frac{u}{v}\) and show the final simplified derivative exactly as done on the teacher’s board.
The Question
Given that:
show that:
Step-by-step working (Teacher’s method)
Setup:
Step 1: Differentiate \(u\)
Step 2: Differentiate \(v\)
Step 3: Apply quotient rule
Quotient rule: \(\displaystyle \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\)
Step 4: Simplify
Since: \(\left((2x+1)^{\frac12}\right)^2=2x+1\)
Combine numerator (same denominator):
Step 5: Final simplification
✅ Hence shown
Key concepts
| Concept | Why it matters |
|---|---|
| Quotient rule | Correct structure for \(y=\frac{u}{v}\) |
| Chain rule | Needed to differentiate \((2x+1)^{\frac12}\) |
| Index laws | Combine powers: \((2x+1)^{\frac12}(2x+1)=(2x+1)^{\frac32}\) |
| Careful algebra | Prevents sign and fraction mistakes |
Tips for students
- Label first: Write \(u\) and \(v\) clearly before differentiating.
- Use chain rule: Power rule and derivative of the inside for \(v=(2x+1)^{\frac12}\).
- Simplify in indices: Express terms like \((2x+1)^{\frac32}\) to keep working clean.
- Keep layout neat: Stick to the quotient rule structure to avoid missing terms.
For parents
Many students lose marks in differentiation not because they “don’t know calculus,” but due to small algebra slips while simplifying (O-Level A-Maths).
At MasterMaths, we train students to (1) set up the method correctly, (2) write clean working, and (3) simplify confidently—exactly what exam markers want to see.