Exercise 3: Proof of Formulas 2 and 3: Derivative of a Function Multiplied by a Vector Function

In this video I prove formulas 2 and 3 of the differentiation rules for vector functions and show that the derivative of a vector function multiplied by a real-valued function has the same result as the product rule. Formula 2 is a special case of Formula 3 in which the function is equal to a constant, thus the derivative of the constant term goes to zero. I prove Formula 3 by taking the derivative by components, which results in applying the product rule multiple times, and then factoring out the real-valued function terms. Then, separating out the terms to obtain two separate vector functions, we obtain our proof!

#math #vectors #calculus #stem #education

Timestamps:

- Exercise 3: Proof of Formulas 2 and 3: 0:00
- Formula 2 is a special case of Formula 3 where f(t) = c: 1:08
- Proof of Formula 3 by taking derivative of components: 2:27
- Factor out f(t) and f'(t) terms and separate into two vector functions: 5:38
- Final result appears similar to the product rule for real valued functions: 6:56

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-co8lcsbc
- Notes: https://peakd.com/hive-128780/@mes/hfsetrue
- Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous
- Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous .

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