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2020 HSC Mathematics Extension 1

Question 13(a):
Differentiation techniques

5 marksdifferentiation techniquesdependency-safe group

(i) Find .

(ii) Use the substitution to evaluate .

How to recognise this question

Question type: differentiation techniques

  • The command and notation point to differentiation techniques.
  • The word “hence” means the later part is intended to reuse the earlier result.

How to handle it: apply product, chain or quotient rules carefully, simplifying only after differentiating.

Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) Chain rule: .
  2. NESA (i): 1 mark for the correct derivative.
  3. (ii) , . Limits: ; .
  4. Integrand: .
  5. Integral becomes by part (i).
  6. .
  7. NESA (ii): 4 marks complete; partial for substitution progress and simplified trig integrand.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 5 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

(i) Differentiate . Hence,

(ii) Use to evaluate .

See the step-by-step answer guide →

Question 2

(i) Find . Hence,

(ii) Evaluate using .

See the step-by-step answer guide →

Question 3

(i) Differentiate . Hence,

(ii) With , show by transforming to .

See the step-by-step answer guide →