easy derivative but it took me 32 minutes

We will find the second derivative of sin(x^2) but without the chain rule, power rule, and product rule. Instead, we will just need ONE limit! This is a hidden gem for limit enthusiasts! The key is to use the limit definition of the second symmetric derivative to find the second derivative of sin(x^2). Note, that if the function is second-differentiable in the usual sense, then the second symmetric derivative is equal to the regular second derivative. Since sin(x^2) is always differentiable, so let's find d^2/dx^2(sin(x^2)) with this rarely-taught approach. We will also need two famous limits, the limit as x goes to 0 of sin(x)/x and the limit as x goes to 0 of (1-cos(x))/x.

0:00 finding the second derivative of sin(x^2) but no differentiation shortcuts
0:54 Definition of derivative, symmetric derivative, and second symmetric derivative in terms of limits
7:46 The real action!!
9:08 The formula for sin(A)+sin(B)
12:01 Back to the limit
29:50 Check the answer!

Check out my 100-limit video for more calculus limit practice.    • 100 calculus limits (ft epsilon-delta...  
----------------------------------------
🛍 Shop my math t-shirt & hoodies: https://amzn.to/3qBeuw6
💪 Get my math notes by becoming a patron:   / blackpenredpen  
#blackpenredpen #math #calculus #apcalculus
----------------------------------------
💪 Support the channel and get featured in the video description by becoming a patron:   / blackpenredpen  
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Samuel N Anthony Rogers Mark Madsen Robert Da Costa Nathan Kean Timothy Raymond Gregory Henzie Lauren Danielle Nadia Rahman Evangline McDonald Yuval Blatt Zahra Parhoun Hassan Alashoor Kaakaopuupod bbaa Joash Hall Andr3w11235 Cadentato Joe Wisniewski Eric Maximilian Mecke Jorge Casanova Alexis Villalobos Jm Law Siang Qi Tancredi Casoli Steven Sea Shanties Nick K Daniel Akheterov Roy Logan
----------------------------------------
Thank you all!