Variation 3 · answer guide

Inverse trigonometric functions — Differentiate an inverse trig function

Question

Differentiate and find .

This practice question was inspired by Question 6 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: use the restricted domain and range that make the inverse function single-valued.Use .
Simplify the previous line carefully, keeping exact values where possible.With , , so .
Finish the calculation, then check that the result meets the question’s conditions.At , .
Match the result to the choices and state the correct option.Therefore option C is correct: and .

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

Verification: A numerical difference quotient near agrees with .

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