AQA AS Level Mathematics Paper 2, June 2025: Question 2

📋 What this question tests

This question assesses your foundational calculus skills at AS Level, specifically:

Understanding functional calculus notation: distinguishing between f(x), f'(x), and f''(x).
Applying the power rule for differentiation: d/dx [a·xⁿ] = a·n·xⁿ⁻¹ .
Recognising the direction of calculus operations (differentiating forward vs. integrating backward).

The second option, 6x , must be clearly circled.

Mark Scheme:
• B1 (AO 1.1b) for circling the second answer ( 6x ).

Identify starting term: We are given the first derivative, f'(x) = 3x² .
Understand the target: f''(x) is the derivative of f'(x) :
f''(x) = d/dx [f'(x)]
Multiply by the current power:
3 × 2 = 6
Decrease power by 1:
2 - 1 = 1 → x¹ = x
Combine:
f''(x) = 6x

Successive Derivatives:
- f(x) → Original function
- f'(x) → First derivative (rate of change / gradient function)
- f''(x) → Second derivative (rate of change of the gradient)
Moving down the chain ( f → f' → f'' ) always means differentiating. Moving up ( f' → f ) requires integrating.

Circling x³ (Integrating instead of differentiating): The most common trap. Students see f'(x) and automatically integrate to find f(x) = ∫ 3x² dx = x³ + c . Always check how many prime marks are in the target expression!
Circling 6 (Differentiating too far): Differentiating 6x gives 6 , which is f'''(x) (the third derivative), not f''(x) .
Circling 1.5x³: Arises from a blend of errors—adding to the power and dividing incorrectly: (3/2)x³ .

Single Clear Mark: If you circle an option and change your mind, cross out the incorrect one cleanly and clearly circle the intended choice. If two options appear circled without clarification, zero marks are awarded.
Watch the Prime Symbols: A single tick mark ' means first derivative; double tick marks '' mean second derivative. Before calculating, ask yourself: "Am I differentiating or integrating?"

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