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

Math Easy Solutions

description

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.

@ 0:00 - Intro
@ 0:42 - Topics to Cover

@ 1:14 - Recap on the Product Rule for Derivatives
@ 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

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