Scott Pauls Department of Mathematics Dartmouth College Math 3, Fall 2015 Scott Pauls Department of Mathematics Dartmouth College Class 9 - 10/5/2015

Seating 11 - the W𝜙ΣS 10 - Don't drink and derive 9 - Apple 𝜋 4 - The Dirty Derivatives 3 - Too cool to function 2 - Gee I'm a tree 1 - Push it to the limit 8 - ∞ and beyond 7 - Gettin' triggy with it 6 - Shapes on a plane 5 - Kool by induction 1 2 3 4 5 6 7 8 9 10 11 Front of the classroom Facilitator Scribe Presenter/Timer Checker Wildcard Wildcard Facilitator Scribe Presenter/Timer Checker

Rules, rules, rules… Sums: 𝑑 𝑑𝑥 𝑓 𝑥 +𝑔 𝑥 = 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑑 𝑑𝑥 𝑔 𝑥 Constants: 𝑑 𝑑𝑥 𝐶=0, 𝑑 𝑑𝑥 𝐶𝑓 𝑥 =𝐶 𝑑 𝑑𝑥 𝑓 𝑥 Power rule: 𝑑 𝑑𝑥 𝑥 𝑛 =𝑛 𝑥 𝑛−1 , 𝑛≠0 𝑑 𝑑𝑥 𝑥 0 =0 Product rule: 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 =𝑔 𝑥 𝑑 𝑑𝑥 𝑓 𝑥 +𝑓 𝑥 𝑑 𝑑𝑥 𝑔(𝑥) Quotient rule: 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑑 𝑑𝑥 𝑓 𝑥 −𝑓 𝑥 𝑑 𝑑𝑥 𝑔(𝑥) 𝑔 𝑥 2 Class 9 - 10/5/2015

Special functions and their derivatives 𝑑 𝑑𝑥 sin 𝑥 = cos 𝑥 𝑑 𝑑𝑥 cos 𝑥 =− sin 𝑥 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥 𝑑 𝑑𝑥 ln 𝑥 = 1 𝑥 Class 9 - 10/5/2015

Finding the quotient rule using the product rule Class 9 - 10/5/2015

Triage problems Find 𝑓′(𝑥) where 𝑓 𝑥 = 𝑥 3 sin⁡(𝑥) 𝑓 𝑥 = ln 𝑥 tan 𝑥 𝑓 𝑥 = −4+ sin 𝑥 𝑥+ cos 𝑥 𝑓 𝑥 = 𝑒 𝑥 xcsc 𝑥 Class 9 - 10/5/2015

Make your own! In groups, make two or three problems which can be solved using the following outlines: Step 1: Quotient rule Step 2: Power rule Step 3: Special differentiation formula B. Step 1: Product rule Step 2: Special differentiation formula C. Create your own set of steps and associated problems. Try to make the problems as hard as possible! Class 9 - 10/5/2015