AQA A-Level Mathematics Paper 1, June 2025: Question 1

1 mark · Easy difficulty · Short Answer

Find the derivative of the exponential curve y = 3e^x.

Practise this question

Question

Question 1 states: 'The equation of a curve is given by y = 3e^x. Find an expression for dy/dx. Circle your answer.' Below are four multiple-choice options arranged horizontally: 3e^(-x), 3e^1, 3e^x, and 3xe^(x-1). The question is worth 1 mark.
Question text

1 The equation of a curve is given by

y = 3ex

dy

Find an expression for

dx

Circle your answer.

[1 mark]

3e–x 3e1 3ex 3xex–1

Mark scheme

Show the mark scheme Mark scheme table for Question 1: Marking instructions state 'Circles third option', AO is 1.2, Marks awarded is B1, and the typical solution is '3e^x'. Total 1 mark.

Q Marking instructions AO Marks Typical solution

1 Circles third option 1.2 B1 3ex

Question 1 Total 1

How to answer it

Differentiation of the Natural Exponential Function y = 3eˣ

📋 What this question tests

This multiple-choice opening question assesses fundamental calculus knowledge under AO1.2 (Routine procedures). Specifically, it tests your ability to differentiate a standard natural exponential function multiplied by a constant scalar: knowing the derivative of eˣ with respect to x is uniquely itself, eˣ , and applying the constant multiple rule.

Question 1 (1 Mark)

Multiple Choice: Find dy/dx when y = 3eˣ

✅ Correct Answer

The correct option to circle is the third option:

3eˣ

Mark Scheme Breakdown:
[B1] (AO 1.2) Circles the third option ( 3eˣ ). Independent mark for the correct identification.

💡 Key Knowledge

  • Fundamental Rule: The derivative of the natural exponential function is invariant:
    d/dx (eˣ) = eˣ
  • Constant Multiple Rule: If k is a constant scalar, then:
    d/dx [k · f(x)] = k · f'(x)
  • Combining these gives:
    d/dx (3eˣ) = 3 · d/dx (eˣ) = 3eˣ

📐 Step-by-Step Calculation

  1. Identify the function form:
    Given y = 3eˣ , the function is in the form y = k · eˣ where k = 3 .
  2. Differentiate with respect to x:
    dy/dx = 3 · (eˣ)' = 3eˣ
  3. Select matching option:
    Options provided: 3e⁻ˣ , 3e¹ , 3eˣ , 3xeˣ⁻¹ .
    Direct match is the 3rd option.

❌ Common Errors & Distractors

  • Applying the Power Rule (Trap!):
    Choosing 3xeˣ⁻¹ comes from incorrectly applying d/dx (xⁿ) = n·xⁿ⁻¹ . Remember: the power rule only works when the base is the variable and the exponent is a constant number, not when the variable is in the exponent!
  • Confusing with linear differentiation:
    Choosing 3e¹ arises from mistakenly treating e as a variable raised to power 1, or erroneously setting x = 1 .
  • Negative power confusion:
    Choosing 3e⁻ˣ confuses standard differentiation with integrating or applying chain rule with negative powers like e⁻ˣ .

🧠 Exam Technique & Examiner Guidance

  • Clear Circling: Ensure your circle clearly encloses only one option. If you change your mind, cross out the incorrect choice cleanly and re-circle your final choice. Ambiguous marks receive 0.
  • Pace Yourself: Question 1 is designed to be completed in under 30 seconds. Do not overcomplicate it or second-guess standard identities.
  • Spot the Variable: Always verify where the independent variable resides before applying differentiation rules:
    • Base = variable ( xⁿ ) → Power Rule
    • Exponent = variable ( aˣ , eˣ ) → Exponential Rule

Topics

Pure Mathematics · G: Differentiation · F: Exponentials and logarithms

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.