Advanced Differential Calculus small Quiz 4-5 quesion in 30min

I need a brief yet challenging quiz focused strictly on advanced differential calculus. Please create three to four questions that can realistically be solved in a 30-minute sitting. Each problem should test deep understanding of differentiation techniques—think implicit differentiation, related rates, higher-order derivatives, or tricky applications of the chain rule—rather than routine textbook drills.

Along with the questions, include fully worked solutions and the final numerical or algebraic answers so I can check grading quickly.

Deliverables:
• 3–4 advanced differential-calculus questions
• Complete, clearly written step-by-step solutions
• Final answers called out for fast marking