Deriving the Quotient Rule from first principles

[@kingbeauregard]

I really like that you aren't frightened of going back to the original definitions.

[@jan-willemreens9010]

... A good day Newton, Nice to see this derivation again after so many years; I can say that nothing has changed between the way of my old math teacher and your way (same math trick); as if time has stood still, but unfortunately this does not apply to my aging proces (lol) ... Take care and thank you Newton, Jan-W

[@holyshit922]

Quotient rule is a part of Ostrogradsky's isolation of rational part of integral We have integral Int(P(x)/Q(x),x) where P(x) and Q(x) are polynomials Int(P(x)/Q(x),x) = P1(x)/Q1(x)+Int(P2(x)/Q2(x),x) P1(x) and P2(x) are unknown polynomials of degree at most one less then degree of their denominators To find this polynomials we use umdetermined coefficients Q1(x) = GCD(Q(x),Q'(x)) Q(x) = Q1(x)Q2(x) How to find GCD ? If you have given explicitly factorization of Q(x) use it If not use long division just like in Euclidean algorithm Int(P(x)/Q(x),x) = P1(x)/Q1(x)+Int(P2(x)/Q2(x),x) P(x)/Q(x) = (P1'(x)Q1(x) - P1(x)Q1'(x))/Q1^2(x) + P2(x)/Q2(x) P(x)/Q(x) = (P1'(x)Q1(x)Q2(x) - P1(x)Q1'(x)Q2(x)+Q1^2(x)P2(x))/(Q1^2(x)Q2(x)) P(x)/Q(x) = (P1'(x)Q1(x)Q2(x) - P1(x)Q1'(x)Q2(x)+Q1^2(x)P2(x))/(Q(x)Q1(x)) P(x)/Q(x) = Q1(x)(P1'(x)Q2(x) - P1(x)(Q1'(x)Q2(x))/Q1(x) + Q1(x)P2(x))/(Q(x)Q1(x)) Let H(x) = Q1'(x)Q2(x)/Q1(x) be an auxiliary polynomial (You can show that H(x) always be a polynomial) P(x)/Q(x) = Q1(x)(P1'(x)Q2(x) - P1(x)H(x) + P2(x)Q1(x))/(Q(x)Q1(x)) P(x)/Q(x) = (P1'(x)Q2(x) - P1(x)H(x) + P2(x)Q1(x))/Q(x) P(x) = P1'(x)Q2(x) - P1(x)H(x) + P2(x)Q1(x) This isolation of rational part of integral is useful when partial fraction decomposition is not so great When numerator has repeated roots especially complex ones

[@shirasayakatana573]

Easy to follow and concise. Thank you!

[@emanuelndawala]

Thank you for great explanation keep up good work sir

[@JosephMorza]

this was really fascinating!

[@itsohaya4096]

Limits are funky!!

[@giovannimezzadri266]

very nice video, you make math more fun. thank you very much

[@johnnolen8338]

Very cool! 😎 (I don't want to steal your thunder but you can get the identical result in three lines using logarithmic differentiation. I never remember the quotient rule ... because I don't have to remember the quotient rule. Oops, I think that was supposed to be a secret. 🤐)

[@PrimeNewtons]

I agree. Sometimes, the teacher requires using the quotient rule. It is why we learn everything 😌

[@GaneshRao-e4j]

Awesome explanation. Other videos just mention the rule. Rarely the derivation is found

[@punditgi]

Prime Newtons rocks 🎸 😊

[@PrimeNewtons]

You too!

[@punditgi]

@@PrimeNewtons 🤩

[@newmomojdid2350]

[@chokoe430]

why do you push h up and how may we supposed to see when we have to push the h up ..... i don't understand why

[@PrimeNewtons]

A proof requires that you know what you want to accomplish. So, you will know what you need because you already know where you're going.

[@chokoe430]

@@PrimeNewtons I'm confused

[@spiffvr9425]

Every manipulation is dove to try to get close to the end result you already known. In this case we move the h up because we known the detention of the derivative involves dividing by h.

[@shirasayakatana573]

You resemble one of my former lecturers. Except they have none of your patient vibe😂

[@kaushik_rebel]

I feel like i remembered my powers back ....