Proof of Chain Rule

Find Textbook Solutions to Stewart Calculus 7th edition, Early Trans. at http://www.prepanywhere.com/category/catalog/collegeuniversity/calculus-7th-e...

A detailed proof of chain rule. If you were ever curious on how the chain rule came about, despite how well you use it, here is the proof. This proof isn't super rigorous using the full delta epsilon proof but I believe it's more easily understandable to the viewers without it. If you have done delta epsilon proof, try to polish this proof up with the delta epsilon proof.

Category:

Education

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• chain rule proof
• calculus
• mat 135
• prepanywhere

[GrandMasterJuan]

i got lost when it came to "b = g(a)", why ais he doing this aswell as "f(b)" .... what do they represent and why are hey not the same ?

[MGLmath]

@GrandMasterJuan y is f composed of g(x). I am first inserting the value of a into g.... and letting b be the output of g. But f takes the output of g as input, hence f(b).

[GrandMasterJuan]

just a question, do they ever teach you about epsilon and delta in calculus, i am in calc 1 right now, and saw those symbols used for the definition of a limit, but don't know where they came about....

[MGLmath]

@GrandMasterJuan delta and epsilon is usually taught in a engineering or math specialist course at the University Level. A typical Calculus course such as the AP courses do that cover delta and epsilon. Delta and Epsilson reasoning is the key to truly understanding what's happening in the world of infinity with functions =)

[GrandMasterJuan]

@MGLmath aaaah thankyou

[Cascaderade]

@johnqwertyful e1/delta x is not indeterminate because epsilon 1 is not 0. It is the small error between the y values of the curve and the tangent line. It becomes 0 because as the change in x approaches 0, then the y value error between the curve and the tangent line gets closer and closer to 0 as well. :)

[MGLminlee]

@johnqwertyful Hey there. The key is in understanding the how we selected epsilon. Note that epsilon is the difference between dy and delta y. And note, how small that difference is compared to delta x. Note in the graph I first introduced in the epsilon how small it is compared to delta x. Hope that helps!

[johnqwertyful]

Near the end I am a bit confused. lim Δx,e1 ->0 e1/Δx is indeterminate. You kind of skipped over WHY it's 0. I know it is 0, but why? If it's indeterminate, it could be 7 or 3 or 900 or pi or whatever.

[AzaLin00]

You kick some serious ass. Keep up the good work.

[kazkillerman]

Great Explanation!