Power Rule Proof, Linearity of Differentiation, Polynomials (Calculus Problems and Solutions)

Use the limit definition of the derivative to prove the power rule for derivatives: (x^n)’=nx^n-1 when n is a positive integer. This is also written d/dx x^n=nx^n-1. Prove (mx+b)’=m as well. Then compute derivatives of power functions and polynomial functions using linearity of differentiation (linearity of the derivative operator). Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

The Binomial Theorem is used with “higher-order terms” here. It’s related to Pascal’s triangle. It can also be written in terms of summations involving combinations (n choose k).

🔴 Calculus 1, Lesson 1.1: Linear Functions (3Blue1Brown SoME2): https://www.youtube.com/watch?v=lmCjnxzIals

🔴 Calculus 1, Lesson 1.2: Exponential Functions: https://www.youtube.com/watch?v=47ZqW19AjD4

🔴 Calculus 1, Lesson 1.3: Quadratic Functions, Transformations of Functions, and Function Composition: https://www.youtube.com/watch?v=1T58CnSME4g

🔴 Calculus 1, Lesson 1.4: Logarithmic Functions (Intro to Horizontal Compressions & Inverse Functions): https://www.youtube.com/watch?v=oS-gvIoJf60

🔴 Calculus 1, Lesson 1.5: Trigonometric Functions and Inverse Trigonometric Functions: https://www.youtube.com/watch?v=UFYjlnGUenE

🔴 AP Calculus AB Exam Review: Practice Exam Problems & Solutions (Multiple Choice, No Calculator): https://www.youtube.com/watch?v=2AG_Dt3x7q0

🔴 AP Calculus AB Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=RLgUwP44V_M

🔴 AP Calculus BC Exam Review: Practice Exam (30 Question Multiple Choice, No Calculator): https://www.youtube.com/watch?v=1Q7ALcwT97A

🔴 AP Calculus BC Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=ZWisJEDP44w

🔴 Calculus 1 Exam 1 Review Problems and Solutions (Precalculus, Limits, Continuity, Derivatives): https://www.youtube.com/watch?v=UjmvUwvd7c4

🔴 Calculus 1 Course, Lecture 1: The Big Ideas (Rates & Areas, the Infinity Principle & Circular Area): https://www.youtube.com/watch?v=2V_ehJC5QUY

🔴 Calculus 1 Course Lecture Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MkFrKtM0S0NLu-TxZa5jBIx

#calculus #PowerRule #linearity

Links and resources
===============================
🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1
🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/
🔴 Follow me on Twitter: https://twitter.com/billkinneymath
🔴 Follow me on Instagram: https://www.instagram.com/billkinneymath/
🔴 You can support me by buying “Infinite Powers, How Calculus Reveals the Secrets of the Universe”, by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA.
🔴 Check out my artist son Tyler Kinney’s website: https://www.tylertkinney.co/

(0:00) Prove the Power Rule when n is a positive integer (with the limit definition of the derivative)
(8:09) Prove (m*x+b)’=m for all x with the limit definition of the derivative
(11:43) Find the derivatives if possible (with the Power Rule and Linearity of the Derivative Operator)
(18:01) Find the derivative (involves fifth root and a fraction) with the Power Rule and Linearity
(21:04) Derivative and tangent line of polynomial function
(27:21) Free fall problem (use derivatives)

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.