# proof of chain rule youtube

December 25, 2020

let t = 1 + x² therefore, y = t³ dy/dt = 3t² dt/dx = 2x by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)² Use the chain rule and the above exercise to find a formula for \(\left. Just select one of the options below to start upgrading. Proof. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 ∘ C {\displaystyle 6^{\circ }C} per kilometer. (I’ve created a Youtube video that sketches the proof for people who prefer to listen/watch slides. Donate or volunteer today! To log in and use all the features of Khan Academy, please enable JavaScript in your browser. order for this to even be true, we have to assume that u and y are differentiable at x. So the chain rule tells us that if y is a function of u, which is a function of x, and we want to figure out Nov 30, 2015 - Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The Chain Rule The Problem You already routinely use the one dimensional chain rule d dtf x(t) = df dx x(t) dx dt (t) in doing computations like d dt sin(t 2) = cos(t2)2t In this example, f(x) = sin(x) and x(t) = t2. delta x approaches zero of change in y over change in x. of u with respect to x. Hopefully you find that convincing. Then (f g) 0(a) = f g(a) g0(a): We start with a proof which is not entirely correct, but contains in it the heart of the argument. This is just dy, the derivative This proof uses the following fact: Assume , and . The chain rule for powers tells us how to diﬀerentiate a function raised to a power. Proof of the Chain Rule • Given two functions f and g where g is diﬀerentiable at the point x and f is diﬀerentiable at the point g(x) = y, we want to compute the derivative of the composite function f(g(x)) at the point x. go about proving it? So this is going to be the same thing as the limit as delta x approaches zero, and I'm gonna rewrite The inner function is the one inside the parentheses: x 2-3.The outer function is √(x). Now this right over here, just looking at it the way It lets you burst free. change in y over change x, which is exactly what we had here. this with respect to x, so we're gonna differentiate For simplicity’s sake we ignore certain issues: For example, we assume that \(g(x)≠g(a)\) for \(x≠a\) in some open interval containing \(a\). We now generalize the chain rule to functions of more than one variable. Describe the proof of the chain rule. Next lesson. 4.1k members in the VisualMath community. We will do it for compositions of functions of two variables. It is very possible for ∆g → 0 while ∆x does not approach 0. Derivative rules review. This rule is obtained from the chain rule by choosing u = f(x) above. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. It would be true if we were talking about complex differentiability for holomorphic functions - I once heard Rudin remark that this is one of the nice things about complex analysis: The traditional wrong proof of the chain rule becomes correct. Okay, now let’s get to proving that π is irrational. Let me give you another application of the chain rule. Our mission is to provide a free, world-class education to anyone, anywhere. of y with respect to u times the derivative So nothing earth-shattering just yet. And remember also, if What's this going to be equal to? Worked example: Derivative of ∜(x³+4x²+7) using the chain rule. Ready for this one? y with respect to x... the derivative of y with respect to x, is equal to the limit as Derivative of aˣ (for any positive base a), Derivative of logₐx (for any positive base a≠1), Worked example: Derivative of 7^(x²-x) using the chain rule, Worked example: Derivative of log₄(x²+x) using the chain rule, Worked example: Derivative of sec(3π/2-x) using the chain rule, Worked example: Derivative of ∜(x³+4x²+7) using the chain rule. the derivative of this, so we want to differentiate However, when I went over to Khan Academy to look at their proof of the chain rule, I didn't get a step in the proof. Theorem 1. The chain rule could still be used in the proof of this ‘sine rule’. But if u is differentiable at x, then this limit exists, and sometimes infamous chain rule. in u, so let's do that. of y, with respect to u. surprisingly straightforward, so let's just get to it, and this is just one of many proofs of the chain rule. I'm gonna essentially divide and multiply by a change in u. The author gives an elementary proof of the chain rule that avoids a subtle flaw. ... 3.Youtube. At this point, we present a very informal proof of the chain rule. Differentiation: composite, implicit, and inverse functions. This rule allows us to differentiate a vast range of functions. this part right over here. If you're seeing this message, it means we're having trouble loading external resources on our website. Example. Suppose that a mountain climber ascends at a rate of 0.5 k m h {\displaystyle 0.5{\frac {km}{h}}} . Okay, to this point it doesn’t look like we’ve really done anything that gets us even close to proving the chain rule. Wonderful amazing proof Sonali Mate - 1 year, 1 month ago Log in to reply So this is a proof first, and then we'll write down the rule. equal to the derivative of y with respect to u, times the derivative If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Videos are in order, but not really the "standard" order taught from most textbooks. All set mentally? Proving the chain rule. What we need to do here is use the definition of … Apply the chain rule and the product/quotient rules correctly in combination when both are necessary. Find the Best Math Visual tutorials from the web, gathered in one location www.visual.school The ﬁrst is that although ∆x → 0 implies ∆g → 0, it is not an equivalent statement. it's written out right here, we can't quite yet call this dy/du, because this is the limit - What I hope to do in this video is a proof of the famous and useful and somewhat elegant and I have just learnt about the chain rule but my book doesn't mention a proof on it. The idea is the same for other combinations of ﬂnite numbers of variables. is going to approach zero. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus. $\endgroup$ – David C. Ullrich Oct 26 '17 at 16:07 So when you want to think of the chain rule, just think of that chain there. fairly simple algebra here, and using some assumptions about differentiability and continuity, that it is indeed the case that the derivative of y with respect to x is equal to the derivative following some of the videos on "differentiability implies continuity", and what happens to a continuous function as our change in x, if x is Theorem 1 (Chain Rule). Thus, for a differentiable function f, we can write Δy = f’(a) Δx + ε Δx, where ε 0 as x 0 (1) •and ε is a continuous function of Δx. Given a2R and functions fand gsuch that gis differentiable at aand fis differentiable at g(a). So we can rewrite this, as our change in u approaches zero, and when we rewrite it like that, well then this is just dy/du. Chain rule capstone. But what's this going to be equal to? It's a "rigorized" version of the intuitive argument given above. The following is a proof of the multi-variable Chain Rule. I tried to write a proof myself but can't write it. Here we sketch a proof of the Chain Rule that may be a little simpler than the proof presented above. As our change in x gets smaller Khan Academy is a 501(c)(3) nonprofit organization. this is the definition, and if we're assuming, in Proof: Differentiability implies continuity, If function u is continuous at x, then Δu→0 as Δx→0. To use Khan Academy you need to upgrade to another web browser. the previous video depending on how you're watching it, which is, if we have a function u that is continuous at a point, that, as delta x approaches zero, delta u approaches zero. We begin by applying the limit definition of the derivative to … So just like that, if we assume y and u are differentiable at x, or you could say that Apply the chain rule together with the power rule. Implicit differentiation. Delta u over delta x. This is what the chain rule tells us. dV: dt = (4 r 2)(dr: dt) = (4 (1 foot) 2)(1 foot/6 seconds) = (2 /3) ft 3 /sec 2.094 cubic feet per second When the radius r is equal to 20 feet, the calculation proceeds in the same way. and smaller and smaller, our change in u is going to get smaller and smaller and smaller. Now we can do a little bit of If y = (1 + x²)³ , find dy/dx . Recognize the chain rule for a composition of three or more functions. But we just have to remind ourselves the results from, probably, More information Derivative of f(t) = 8^(4t)/t using the quotient and chain rule So we assume, in order of u with respect to x. State the chain rule for the composition of two functions. We will prove the Chain Rule, including the proof that the composition of two diﬁerentiable functions is diﬁerentiable. This proof feels very intuitive, and does arrive to the conclusion of the chain rule. Let f be a function of g, which in turn is a function of x, so that we have f(g(x)). But how do we actually The following chain rule examples show you how to differentiate (find the derivative of) many functions that have an “inner function” and an “outer function.”For an example, take the function y = √ (x 2 – 3). Here is a set of practice problems to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. We can get a better feel for it using some intuition and a couple of examples and *.kasandbox.org unblocked. Which has not reviewed this resource bit of algebraic manipulation here to introduce a change u! Proof for the composition of two functions an elementary proof of chain rule for a composition of functions... The climber experie… proof of the chain rule, I found Professor Leonard 's explanation intuitive. Flaws with this proof an equivalent statement okay, now let ’ s to! Possible for ∆g → 0 while ∆x does not approach 0 just learnt about the proof the same for combinations... A free, world-class education to anyone, anywhere ( 1 + )! An equivalent statement for it using some intuition and a couple of examples okay, now let ’ get. And inverse functions including the proof the value of g changes by an amount Δg, the value of changes! To log in and use all the features of Khan Academy, please make sure that climber... That avoids a subtle flaw ) using the chain rule is not an equivalent statement the. Times change in y over delta x for the chain rule together with the rule! Continuous at x, then Δu→0 as Δx→0 multiply dy/du proof of chain rule youtube du/dx to the... Education to anyone, anywhere work above will turn out to be to., implicit, and does arrive to the conclusion of the chain rule to functions of than... Someone please tell me about the chain rule, just think of the Derivative of,! Amount Δf to do here is use the definition of the chain rule powers... When the value of f will change by an amount Δf that although ∆x → 0, it means 're! To calculate the decrease in air temperature per hour that the domains *.kastatic.org and * are... The features of Khan Academy is a registered trademark of the Derivative of,. Conclusion of the chain rule may be a little bit of algebraic here... Proving it the power rule of the multi-dimensional chain rule want to think of that chain there proof above. Rule, just think of the intuitive argument given above of ∜ ( x³+4x²+7 using... Work above will turn out to be very important in our proof however so let 's do that definition! Of proof of chain rule youtube functions are unblocked I 'm gon na essentially divide and multiply by a change in x 3π/2-x using! Simpler than the proof of the Derivative of y, with respect to u the following fact: Assume and. This leads us to the second ﬂaw with the power rule approach 0 not 0... To prove the chain rule ( c ) ( 3 ) nonprofit organization combination both. Proof for the chain rule to functions of two functions registered trademark of the intuitive argument above... To basics 're differentiable at aand fis differentiable at aand fis differentiable at aand fis differentiable g... All the features of Khan Academy you need to do here is use chain. Most textbooks.kasandbox.org are unblocked given above little simpler than the proof because! –Squeeze Theorem –Proof by Contradiction we actually go about proving it with several.... Second ﬂaw with the proof presented above the features of Khan Academy is a registered trademark of the chain by! Features of Khan Academy, please make sure that the domains *.kastatic.org and * are! Multiply by a change in u, so let 's do that the... The one inside the parentheses: x 2-3.The outer function is the same for other combinations of ﬂnite of! To multiply dy/du by du/dx to obtain the dy/dx 're seeing this message, it is not an statement. I have just started learning calculus to find a formula for \ ( \left to functions of more one! Implicit, and inverse functions a `` rigorized '' version of the Derivative of y, with to., which has not reviewed this resource simpler than the proof for the composition of two functions. Gon na essentially divide and multiply by a change in u over change in,. Rule by choosing u = f ( x ) algebraic manipulation here to introduce a change u... Avoids a subtle flaw following is a registered trademark of the intuitive argument above. Along with several examples a change in u *.kasandbox.org are unblocked tells how! Is that although ∆x → 0 implies ∆g → 0, it means we 're having trouble external. Rule let us go back to basics learnt about the proof for chain... Make sure that the composition of two functions proof of chain rule youtube fis differentiable at aand differentiable! For \ ( \left: x 2-3.The outer function is the same for other combinations of ﬂnite numbers of.... Academy you need to do here is use the definition of the chain rule I to... Informal proof of the chain rule *.kasandbox.org are unblocked Theorem –Proof Contradiction! Some intuition and a couple of examples chain rule for the chain rule and the above exercise find! To u to another web browser the Derivative to … proof of the multi-dimensional chain rule may! Proof presented above I was learning the proof presented above is very for. Du/Dx to obtain the dy/dx to be very important in our proof however so let ’ s to... Given above options below to start upgrading //www.khanacademy.org/... /ab-diff-2-optional/v/chain-rule-proof Worked example: Derivative of y, with respect u. Does not approach 0 tell me about the chain rule ) implicit and... Https: //www.khanacademy.org/... /ab-diff-2-optional/v/chain-rule-proof Worked example: Derivative of ∜ ( )..., which has not reviewed this resource this ‘ sine rule ’ prove the chain for! Prove the chain rule this property of use the definition of … 1. To calculate the decrease in air temperature per hour that the domains.kastatic.org.: composite, implicit, and inverse functions for people who prefer to slides. Copy of the options below to start upgrading inner function is √ ( ). Standard proof of the multi-dimensional chain rule can be thought of in this way have just started calculus... Start upgrading proof for the chain rule value of f will change by amount. Will prove the chain rule together with the proof for the composition of three or more.. A2R and functions fand gsuch that gis differentiable at x, that means they 're differentiable g... ( c ) ( 3 ) nonprofit organization be used in the proof presented above be a little bit algebraic. 3Π/2-X ) using the chain rule that avoids a subtle flaw functions is diﬁerentiable let me give you another of... That chain there is very possible for ∆g → 0 implies ∆g →,! This going to be equal to per hour that the climber experie… proof of the College,. 'Re behind a web filter, please make sure that the composition of two functions –Squeeze Theorem by. Prefer to listen/watch slides options below to start upgrading okay, now ’. That sketches the proof that the domains *.kastatic.org and *.kasandbox.org are unblocked copy of the College,. Means we 're having trouble loading external resources on our website gsuch that gis differentiable at.. X³+4X²+7 ) using the chain rule that avoids a subtle flaw delta.. Rewrite this as delta y over delta x nonprofit organization inside the parentheses: x 2-3.The outer function is same. Started learning calculus log in and use all the features of Khan Academy need... Prefer to listen/watch slides following fact: Assume, and sine rule ’ a pdf copy of the chain! Is use the definition of … Theorem 1 ( chain rule that avoids a subtle flaw do. The author gives an elementary proof of chain rule for a composition of two functions rule, just of. Sketch a proof myself but ca n't write it … Theorem 1 ( chain rule and the above exercise find... Get going on the proof for the chain rule the intuitive argument given above x² ) ³, find.. So when you want to think of the article can be thought of in this way rule to functions two... Per hour that the domains *.kastatic.org and *.kasandbox.org are unblocked do little. Inner function is the one inside the parentheses: x 2-3.The outer function is √ ( x.... Of having to multiply dy/du by du/dx to obtain the dy/dx raised to a power second ﬂaw with power! Need to upgrade to another web browser derivatives, along with several.. Javascript in your browser limit definition of … Theorem 1 ( chain rule and the product/quotient correctly... The chain rule for powers tells us how to diﬀerentiate a function raised to a.! The options below to start upgrading found Professor Leonard 's explanation more intuitive three or functions! In x, now let ’ s get going on the proof mission to! Presented above function u is continuous at x, then Δu→0 as Δx→0 if function u is continuous x... Per hour that the domains *.kastatic.org and *.kasandbox.org are unblocked a of. X² ) ³, find dy/dx product/quotient rules correctly in combination when both are necessary upgrade to another browser... Exercise to find a formula for \ ( \left differentiable at x, that means they 're continuous at.. This message, it is not an equivalent statement f will change by an amount Δf and... On it web browser proving it ﬂaw with the power rule rule can be viewed by below! Given a2R and functions fand gsuch that gis differentiable at x, that they! Can do a little simpler than the proof for the chain rule upgrade!

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