Differentiation techniques — Upper limit √3

Question

(i) Find . Hence,

(ii) Evaluate using .

This practice question was inspired by Question 13(a) in the 2020 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: apply product, chain or quotient rules carefully, simplifying only after differentiating.(i) .
State the final answer clearly in the form the question requested.Using part (i), (ii) Limits to ; integral .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: , times is .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.