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Leibniz’s Rule: Generalization of the Product Rule for Derivatives

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First published at 02:11 UTC on August 11th, 2020.

#mathematics
#calculus
#science

Math Easy Solutions

MES

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In this video I go over Leibniz’s rule which is the generalization of the product rule for derivatives and extends it for all derivatives and not just the first one. I had covered the product rule in my earlier video and showed that the derivative of the product (f·g)’ = f’g + fg’. We can use this same product rule, along with mathematical induction, to prove the general Leibniz rule for the n-th derivative of a product (f·g)^(n).

Leibniz’s rule makes use of the binomial coefficient which I had briefly discussed in a prior video as well. After I prove the rule, I manually derive consecutive derivatives using the product rule, as well as comparing the result with the binomial theorem. I do this to show the amazing synchronicity or similarity between the powers of a binomial (x + y)^n and the derivative of a product (f·g)^(n).

The topics covered in this video are listed below with their time stamps.

- @ 0:42 - Topics to Cover
1. @ 1:14 - Recap on the Product Rule for Derivatives
2. @ 2:51 - Leibniz's Rule: Generalization of the Product Rule for Derivatives
- @ 8:00 - Proof of Leibniz's Rule
- @ 45:17 - Manually Determining the n-th Derivative Using the Product Rule
- @ 59:27 - Synchronicity with the Binomial Theorem

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiLYb3IimifHz3HfzeQ?e=rch7UF

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/leibniz-general-product-rule-for-derivatives

Related Videos:

Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk

Problems Plus Example 4 - Mathematical Induction: http://youtu.be/WdIr_onvUtE

Infinite Sequences and Series: Representations of Functions as Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series .

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