Inverse function — Second derivative of an inverse

Question

At , suppose , , . Find .

This practice question was inspired by Question 11(a) in the 2025 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: replace the function value by y, swap x and y, then solve for y.Use .
Simplify the previous line carefully, keeping exact values where possible.Here because .
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Final answer:

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

Verification: The denominator is cubed, not squared; .

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