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

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. @ 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

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[mes]

In this video I expand upon the product rule for derivatives, which was for just 1 derivative, and show that it can be generalized to the n-th derivative and which is called the general Leibniz rule.

[codecampbase]

Probably the best video I’ve seen explaining a complete solution.
You do exactly explain the points that raise questions and you explain it in such a way, so that it’s easy to follow!

[rizkaapriliani]

hi sir, thank you very much for the video. it really helps! but i was wondering if you perhaps have any video covering the proof of the generalization of the quotient rule? i've been looking everywhere but couldn't seem to find any video about that