08 09 A dive into the paradoxical nature of the Chain rule Richard Xu The Chain rule, as many of us know, is just the formula “dy/du * du/dx”, a simple way for us to differentiate a composite function. On the surface it appears quite logical, as the “du” part of both derivatives simply cross cancel and we are left with the fraction of “dy/dx”. For a while, that’s what I personally believed as well, that this was just a simple rule and concisely proves the feasibility of the Chain rule. However, if we take a moment to think about this conjecture more deeply, we realise that derivatives aren’t fractions. For example, “dy/dx” isn’t really a fraction but instead “d/dx *(y)” and when deconstructed, means “y” differentiated with respect to “x” or a more intuitive definition: the instantaneous rate of change of “y” with respect to “x” and therefore illustrating that “d/dx” is a notation. Why does the chain rule work then? Brief run through of how the Chain rule works: OR A more intuitive understanding of chain rule is if we imagine these functions in a real-life scenario. Let’s say in a race Adam is 3 times faster than Belial and Belial is 5 times faster than Charlie. Then it can be concluded that Adam is 15 times faster than Charlie. This example could then be distilled down into this: Delving into the origin of differential calculus and the chain rule: There are major debates about whether calculus was discovered by either Newton or Leibniz. However, since our focus is the chain rule, I will explore more of Leibniz’ notation of calculus but feel free to research more about this controversial debate more on your own. When Leibniz first came up with his notation of differentiation, he believed that “dy/ dx” was in fact a quotient or more commonly known as fractions. This was due to how he defined differentiation as an infinitesimal (infinitely small number) change in value of “y”, caused by an infinitesimal change in value of “x”, divided by an infinitesimal change in value of “x”. Visualized: Then, based on this idea, it would be natural for the idea of chain rule to work as both derivatives “dy/du” and “du/dx” are in fact fractions and can therefore cross cancel. Leibniz’ definition of differentiation of course makes intuitive sense but if we were to consider the actual definition of differentiation: the instantaneous rate of change of “y” with respect to “x” suggests that differentiation allows us to find the rate of change at any single point. This is where Leibniz’ definition falls apart as no matter how infinitely small the distance between two points, is there still is, a distance. However, there is a way to make Leibniz’ definition of differentiation work which I will go through later. The idea of limits and why that affects the chain rule: To get over the problems we face with the idea of infinitesimal changes, I will introduce the idea of limits which is a fundamental concept in calculus. I’ll approach this from a more intuitive point of view. Imagine, a frog at the centre of a pond which hops exactly half of the distance between itself and the edge of the pond. The frog then hops half of the distance between its new position and the edge of the pond. Now the frog keeps repeating this process forever, meaning that although the frog will never reach the edge of the pond but it will always be closer than any number we propose. A Dive into the paradoxical nature of the Chain rule The Chain rule as many of us knows it, is just the formula “dy/du * du/dx”, a simple way for us to differen<ate a composite func<on. On the surface it appears quite logical as the “du” part of both deriva<ves simply cross cancel and we are leD with the frac<on of “dy/dx”. For a while that’s what I personally believed as well, that this was just a simple rule and concisely proves the feasibility of the Chain rule. However, if we take a moment to think about this conjecture more deeply, we realise that deriva<ves aren’t frac<ons. For example, “dy/dx” isn’t really a frac<on but instead “d/dx *(y)” and when deconstructed means “y” differen<ated with respect to “x” or a more intui<ve defini<on: the instantaneous rate of change of “y” with respect to “x” and therefore illustra<ng that “d/dx” is a nota<on. Why does the chain rule work then? Brief run through of how the Chain rule works: = $ ( )) = ( ) = ( ) = × OR = ′( ) × ′( ( )) Example: = sin ( !) = 2 ( !) R A more intui<ve understanding of chain rule is if we imagine these func<ons in a real-life scenario. Let’s say in a race Adam is 3 <mes faster than Belial and Belial is 5 <mes faster than Charlie. Then it can be concluded that Adam is 15 <mes faster than Charlie. This example could then be dis<lled down into this: ℎ = × ℎ = 3 × 5 = 15 Aside: The nota.on “ ’ ” is pronounced prime and just means the deriva.ve of that func.on Delving into the origin of diff ren<al calc lus and the chain rule: There are major debates about whether calculus was discovered by either Newton or Leibniz. However, since our focus is the chain rule, I personally will explore more of Leibniz’ nota<on of calculus but feel free to research more about this controversial debate more on your own. When Leibniz first came up with his nota<on of differen<a<on, he believed that “dy/dx” was in fact a quo<ent or more commonly known as frac<ons. This was due to how he defined differen<a<on as an infinitesimal (infinitely small number) change in value of “y” caused by and infinitesimal change in value of “x” divided by an infinitesimal change in value of “x”. Visualized: Then based on this idea it would be natur l fo the idea of chain rule to work as both deriva<ves “dy/du” and “du/dx” are in fact frac<ons and can therefore cross cancel. Leibniz’ defini<on of differen<a<on f c urse makes in ui<ve sense but if w were to c nsider th actual d fini<on f differen<a<on: the instantaneous rate f change o “y” with respect to “x” suggests that differen<a<on allows us to find the rate of change at any single point. This is where Leibniz’ defini<on falls apart as no maZer how infinitely small the distance between two points is there s<ll is a distanc . However, there is a way to make Leibniz’ defini< n of differen<a<on work which I will go through later. The idea of limits and why that affects the chain rule: To get over the problems we face with the idea of infinitesimal changes I will introduce the idea of limits which is a fundamental concept in c lculus. I’ll appro ch this from a more intui<ve point of view. Imagine, a frog at the centre of a pond which hops exactly half of the distance between itself and the edge of the pond. The frog then hops half of the distance between its new posi<on and the edge of t pond. Now the frog keeps repea<ng t is process forev r meaning that although the frog will never reach the edge of the pond but it will always be cl ser than any number we propose. “dx” or an infinitesimal change in value of x “dy” or an infinitesimal change in value of y Frog Delving int the origin of differen<al calculus and the chain rule: There re m jor debates about whether calculus was discovered by either Newton or Leibniz. However, sinc our focus is the chain rule, I p rsonally will explore more of Leibniz’ nota<on of calculus but feel free to research more about this controversial debate more on your own. When Leibniz first came up with his nota<on of differen<a<on, he believed that “dy/dx” was in fact a quo<ent or more commonly known as frac<ons. Th s was due to how he defi ed differen<a<on as an infinitesimal (infinitely small n mber) change in value of “y” caused by and infinitesimal change in value of “x” divided by an infinitesimal change in value of “x”. Visualized: Then based on this idea it would be natural for the idea of chain rule to work as both deriva<ves “dy/du” and “du/dx” are in fact frac<ons and can therefore cross cancel. Leibniz’ defini<on of differen<a<on of course makes intui<ve sense but if we were to consider the actual defini<on of differen<a<on: the instantaneous rate of change of “y” with respect to “x” suggests that differen<a< n allows us to find the rate of change at any single point. This is where Leibniz’ defini<on falls apart as no maZer how infinitely small the distance between two points is there s<ll is a distance. However, there is a way to make Leibniz’ defini<on of differen<a<on work which I will go through later. The idea of limits and why that affects the chain rule: To get over the problems we face with the idea of infinitesimal changes I will introduce the idea of limits which is a fundamental concept in calculus. I’ll approach this from a more intui<ve point of view. Imagine, a frog at the centre of a pond which hops exactly half of the distance between itself and th edg f the pond. The frog then hops half of the distance b tween its n w posi<o and the edge of the pond. Now the fr g keeps repea<ng this process forever meaning that although the frog will never reach the edge of the pond but it will always be closer than any number we propose. “dx” or an infinit si l change in value of x “dy” or an infinitesimal change in value of y Frog A Dive into the paradoxical nature of the Chain rule The Chain rule as many of us knows it, is just the formula “dy/du * d /dx”, a simpl way for us to differen<ate a composi e func<on. On the surfac it appears quit logical as the “du” part of both deriva<v s simply cross cancel a d we are leD with th frac<on of “dy/dx”. For a while that’s what I personally believ d s we l, that is was just a simple rul and concis ly pr ves the feasibility of the Chain rule. How ver, if we t ke a moment to th nk about this conjectur more deepl , we realise that deriva<ves aren’t frac<ons. For example, “dy/ x” is ’t eally a frac<on but instead “d/dx *(y)” and wh n deconstructed m a s “y” differe <ated wit r sp ct to “x” or a more intui<ve defini<on: th instantaneous rate of change of “y” with respect to “x” and therefore illustra<ng that “d/dx” is a nota<on. Why does the chain rule work then? Brief run through of how the Chain rule works: = $ ( )) = ( ) = ( ) = × OR = ′( ) × ′( ( )) Example: = sin ( !) = 2 ( !) OR A more intui<ve understan ing of chain rule is if we imagine thes func<ons in a real-life scenario. Let’s say in a race Adam is 3 <mes faster than Belial and Belial is 5 <mes faster than Charlie. Then it can be concluded that A am is 15 <mes faster than Charlie. This example could then be dis<lled down into this: ℎ = × ℎ = 3 × 5 = 15 Aside: Th nota.on “ ’ ” is pronounced prime and just means the deriva.ve of that func.on
RkJQdWJsaXNoZXIy ODA2Njk=