If y = (2x + 1)$^3$ find $\frac{\text{d}}{\text{dx}}$

Analyze the question: The question is incomplete as typed. It should read "find dy/dx". We differentiate y = (2x + 1)³ with respect to x.

Concept: the Chain Rule If y = [f(x)]ⁿ, then dy/dx = n[f(x)]ⁿ⁻¹ × f′(x). In other words: differentiate the outside (the power), then multiply by the derivative of the inside.

Solution:

1. Let u = 2x + 1, so y = u³.
2. Differentiate the outside: dy/du = 3u² = 3(2x + 1)²
3. Differentiate the inside: du/dx = 2
4. Multiply:

dy/dx = dy/du × du/dx = 3(2x + 1)² × 2 = 6(2x + 1)²

Checking the options:

• A. 3(2x+1)²: you forgot to multiply by 2, the derivative of the inside.
• B. 6(2x+1): wrong power.
• C. 3(2x+1): wrong power and missing the factor of 2.
• D. 6(2x+1)²: matches our working.

Exam tip: Whenever the bracket contains more than just "x", remember to multiply by the derivative of what is inside the bracket. Students often lose marks by leaving out this factor, which is how option A is set as a trap. Quick check: the power drops from 3 to 2, and the coefficient is 3 × 2 = 6.