title
Visualizing the chain rule and product rule | Chapter 4, Essence of calculus

description
A visual explanation of what the chain rule and product rule are, and why they are true. Help fund future projects: https://www.patreon.com/3blue1brown This video was sponsored by Brilliant: https://brilliant.org/3b1b An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/lessons/chain-rule-and-product-rule#thanks Home page: https://www.3blue1brown.com Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Timestamps: 0:00 - Intro 1:48 - Sum rule 4:13 - Product rule 8:41 - Chain rule 14:36 - Outro ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

detail
{'title': 'Visualizing the chain rule and product rule | Chapter 4, Essence of calculus', 'heatmap': [{'end': 213.307, 'start': 189.649, 'weight': 0.706}, {'end': 404.618, 'start': 389.284, 'weight': 0.713}, {'end': 689.677, 'start': 617.003, 'weight': 0.729}], 'summary': "Covers understanding derivatives of complex functions through combining functions and explains the concept of derivatives using the example of f equals sine of x plus x squared, emphasizing its application in finding the total change in height. it also visualizes the product of two functions as the area of a box and discusses the relationship between infinitesimal changes in functions and the derivative, introducing the mnemonic 'left d right, right d left', and demonstrating function composition through examples and visualizations. additionally, it covers the concept of derivatives using the example of nudging x by dx, demonstrates how to calculate the derivative of sine and x squared, and explains the chain rule for derivatives.", 'chapters': [{'end': 259.021, 'segs': [{'end': 33.936, 'src': 'embed', 'start': 1.141, 'weight': 0, 'content': [{'end': 18.404, 'text': 'you In the last videos, I talked about the derivatives of simple functions.', 'start': 1.141, 'duration': 17.263}, {'end': 25.75, 'text': 'And the goal was to have a clear picture or intuition to hold in your mind that actually explains where these formulas come from.', 'start': 19.025, 'duration': 6.725}, {'end': 27.451, 'text': 'But of course,', 'start': 26.71, 'duration': 0.741}, {'end': 33.936, 'text': 'most of the functions you deal with in modeling the world involve somehow mixing or combining or tweaking these simple functions in some other way.', 'start': 27.451, 'duration': 6.485}], 'summary': 'Derivatives of simple functions explained for modeling complex functions.', 'duration': 32.795, 'max_score': 1.141, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA1141.jpg'}, {'end': 105.13, 'src': 'embed', 'start': 75.305, 'weight': 2, 'content': [{'end': 76.665, 'text': 'and then multiplying the two together.', 'start': 75.305, 'duration': 1.36}, {'end': 83.469, 'text': 'So really most functions you come across just involve layering together these three different types of combinations,', 'start': 77.586, 'duration': 5.883}, {'end': 86.351, 'text': "though there's not really a bound on how monstrous things can become.", 'start': 83.469, 'duration': 2.882}, {'end': 91.314, 'text': 'But as long as you know how derivatives play with just those three combination types,', 'start': 87.152, 'duration': 4.162}, {'end': 96.657, 'text': "you'll always be able to just take it step by step and peel through the layers for any kind of monstrous expression.", 'start': 91.314, 'duration': 5.343}, {'end': 105.13, 'text': 'So the question is if you know the derivative of two functions, what is the derivative of their sum,', 'start': 99.062, 'duration': 6.068}], 'summary': 'Functions can be combined in three types, no limit on complexity. understanding derivatives of these types enables peeling through layers for any expression.', 'duration': 29.825, 'max_score': 75.305, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA75305.jpg'}, {'end': 141.278, 'src': 'embed', 'start': 114.839, 'weight': 1, 'content': [{'end': 118.68, 'text': 'The derivative of a sum of two functions is the sum of their derivatives.', 'start': 114.839, 'duration': 3.841}, {'end': 127.802, 'text': "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions,", 'start': 119.76, 'duration': 8.042}, {'end': 135.544, 'text': "since the derivative patterns for products and for function composition won't be so straightforward and they're going to require this kind of deeper thinking.", 'start': 127.802, 'duration': 7.742}, {'end': 141.278, 'text': "For example, let's think about this function f equals sine of x plus x squared.", 'start': 136.636, 'duration': 4.642}], 'summary': 'The derivative of a sum of two functions is the sum of their derivatives, as demonstrated with the example of f(x) = sin(x) + x^2.', 'duration': 26.439, 'max_score': 114.839, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA114839.jpg'}, {'end': 222.678, 'src': 'heatmap', 'start': 189.649, 'weight': 0.706, 'content': [{'end': 195.254, 'text': 'the change to the sine graph is what we might call d sine of x plus.', 'start': 189.649, 'duration': 5.605}, {'end': 198.697, 'text': 'whatever the change to x squared is dx squared.', 'start': 195.254, 'duration': 3.443}, {'end': 207.48, 'text': 'Now we know that the derivative of sine is cosine, and remember what that means.', 'start': 202.714, 'duration': 4.766}, {'end': 213.307, 'text': 'It means that this little change d sine of x is about cosine of x times dx.', 'start': 207.92, 'duration': 5.387}, {'end': 222.678, 'text': "It's proportional to the size of our initial nudge dx, and the proportionality constant equals cosine of whatever input we happen to start at.", 'start': 214.148, 'duration': 8.53}], 'summary': 'The derivative of sine is cosine, indicating that d sine of x is proportional to cosine of x times dx.', 'duration': 33.029, 'max_score': 189.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA189649.jpg'}, {'end': 259.021, 'src': 'embed', 'start': 214.148, 'weight': 3, 'content': [{'end': 222.678, 'text': "It's proportional to the size of our initial nudge dx, and the proportionality constant equals cosine of whatever input we happen to start at.", 'start': 214.148, 'duration': 8.53}, {'end': 227.527, 'text': 'Likewise, because the derivative of x squared is 2x,', 'start': 223.925, 'duration': 3.602}, {'end': 233.709, 'text': 'the change in the height of the x squared graph is going to be about 2 times x times whatever dx was.', 'start': 227.527, 'duration': 6.182}, {'end': 244.053, 'text': 'So rearranging df divided by dx, the ratio of the tiny change to this sum function to the tiny change in x that caused.', 'start': 235.65, 'duration': 8.403}, {'end': 249.976, 'text': 'it is indeed cosine of x plus 2x, the sum of the derivatives of its parts.', 'start': 244.053, 'duration': 5.923}, {'end': 254.338, 'text': 'But like I said, things are a bit different for products.', 'start': 251.496, 'duration': 2.842}, {'end': 256.298, 'text': "And let's think through why.", 'start': 254.818, 'duration': 1.48}, {'end': 259.021, 'text': "And let's think through why in terms of tiny nudges again.", 'start': 256.579, 'duration': 2.442}], 'summary': 'The derivative of a function is proportional to the change in input, with a constant equal to the cosine of the input. for x squared, the change in height is approximately 2 times x times the change in input. the derivative of a sum function is the sum of the derivatives of its parts, while the derivative of a product is different.', 'duration': 44.873, 'max_score': 214.148, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA214148.jpg'}], 'start': 1.141, 'title': 'Derivatives and their understanding', 'summary': 'Covers understanding derivatives of complex functions through three basic ways of combining functions, and explains the concept of derivatives using the example of f equals sine of x plus x squared, emphasizing its application in finding the total change in height.', 'chapters': [{'end': 135.544, 'start': 1.141, 'title': 'Understanding derivatives of complex functions', 'summary': 'Explains how to take derivatives of more complicated function combinations by understanding three basic ways to combine functions: addition, multiplication, and composition, and emphasizes the importance of having a clear picture in mind for where each derivative comes from.', 'duration': 134.403, 'highlights': ['The chapter emphasizes the importance of having a clear picture in mind for where each derivative comes from', 'The chapter explains the three basic ways to combine functions: addition, multiplication, and composition', 'The chapter discusses the derivative patterns for products and function composition, which require deeper thinking']}, {'end': 259.021, 'start': 136.636, 'title': 'Understanding the derivative', 'summary': 'Explains the concept of derivatives using the example of the function f equals sine of x plus x squared, demonstrating how the derivative is calculated and its application in finding the total change in height.', 'duration': 122.385, 'highlights': ['The derivative of sine is cosine, and the change d sine of x is about cosine of x times dx, proportional to the size of the initial nudge dx, and the proportionality constant equals cosine of the input.', 'The derivative of x squared is 2x, and the change in the height of the x squared graph is about 2 times x times dx.', 'The ratio of the tiny change to the sum function to the tiny change in x is indeed cosine of x plus 2x, the sum of the derivatives of its parts.']}], 'duration': 257.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA1141.jpg', 'highlights': ['The chapter emphasizes the importance of having a clear picture in mind for where each derivative comes from', 'The chapter discusses the derivative patterns for products and function composition, which require deeper thinking', 'The chapter explains the three basic ways to combine functions: addition, multiplication, and composition', 'The ratio of the tiny change to the sum function to the tiny change in x is indeed cosine of x plus 2x, the sum of the derivatives of its parts', 'The derivative of sine is cosine, and the change d sine of x is about cosine of x times dx, proportional to the size of the initial nudge dx, and the proportionality constant equals cosine of the input', 'The derivative of x squared is 2x, and the change in the height of the x squared graph is about 2 times x times dx']}, {'end': 592.346, 'segs': [{'end': 289.063, 'src': 'embed', 'start': 260.062, 'weight': 0, 'content': [{'end': 263.043, 'text': "In this case, I don't think graphs are our best bet for visualizing things.", 'start': 260.062, 'duration': 2.981}, {'end': 266.426, 'text': 'Pretty commonly in math at a lot of levels of math.', 'start': 263.784, 'duration': 2.642}, {'end': 272.169, 'text': "really, if you're dealing with a product of two things, it helps to understand it as some kind of area.", 'start': 266.426, 'duration': 5.743}, {'end': 279.094, 'text': 'In this case, maybe you try to configure some mental setup of a box where the side lengths are sine of x and x squared.', 'start': 273.07, 'duration': 6.024}, {'end': 280.935, 'text': 'But what would that mean?', 'start': 279.994, 'duration': 0.941}, {'end': 289.063, 'text': 'Well, since these are functions, you might think of those sides as adjustable, dependent on the value of x,', 'start': 282.281, 'duration': 6.782}], 'summary': 'Graphs not best for visualizing, consider area as product, sine of x and x squared as side lengths.', 'duration': 29.001, 'max_score': 260.062, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA260062.jpg'}, {'end': 325.561, 'src': 'embed', 'start': 301.075, 'weight': 1, 'content': [{'end': 308.717, 'text': 'As you change this value of x up from 0, it increases up to a length of 1 as sin moves up towards its peak.', 'start': 301.075, 'duration': 7.642}, {'end': 315.138, 'text': 'And after that, it starts to decrease as sin comes down from 1.', 'start': 309.517, 'duration': 5.621}, {'end': 318.599, 'text': 'And in the same way, that height there is always changing as x².', 'start': 315.138, 'duration': 3.461}, {'end': 325.561, 'text': 'So f defined as the product of these two functions, is going to be the area of this box.', 'start': 320.04, 'duration': 5.521}], 'summary': 'As x increases, sin(x) increases to 1, then decreases. the height changes as x². the product of these functions gives the area of the box.', 'duration': 24.486, 'max_score': 301.075, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA301075.jpg'}, {'end': 421.551, 'src': 'heatmap', 'start': 389.284, 'weight': 2, 'content': [{'end': 395.77, 'text': "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so that we can actually see them.", 'start': 389.284, 'duration': 6.486}, {'end': 404.618, 'text': 'But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.', 'start': 396.33, 'duration': 8.288}, {'end': 415.648, 'text': 'So applying what we know about the derivative of sine and of x squared, that tiny change dx squared is going to be about 2x times dx.', 'start': 406.383, 'duration': 9.265}, {'end': 421.551, 'text': "And that tiny change d sine of x, well that's going to be about cosine of x times dx.", 'start': 416.308, 'duration': 5.243}], 'summary': 'Using small changes dx, the derivative of sine and x squared can be approximated by 2x*dx and cosine(x)*dx, respectively.', 'duration': 25.221, 'max_score': 389.284, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA389284.jpg'}, {'end': 472.652, 'src': 'embed', 'start': 446.929, 'weight': 3, 'content': [{'end': 451.453, 'text': 'And sometimes people like to remember this pattern with a certain mnemonic that you kind of sing in your head.', 'start': 446.929, 'duration': 4.524}, {'end': 453.675, 'text': 'Left d right, right d left.', 'start': 452.154, 'duration': 1.521}, {'end': 461.882, 'text': 'In this example, where we have sine of x times, x squared left d, right means you take that left function sine of x times,', 'start': 454.536, 'duration': 7.346}, {'end': 464.685, 'text': 'the derivative of the right, in this case 2x.', 'start': 461.882, 'duration': 2.803}, {'end': 466.987, 'text': 'Then you add on right d left.', 'start': 465.425, 'duration': 1.562}, {'end': 472.652, 'text': 'That right function, x squared, times the derivative of the left one, cosine of x.', 'start': 467.527, 'duration': 5.125}], 'summary': "Teach mnemonic 'left d right, right d left' for taking derivatives.", 'duration': 25.723, 'max_score': 446.929, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA446929.jpg'}, {'end': 519.544, 'src': 'embed', 'start': 492.193, 'weight': 4, 'content': [{'end': 495.255, 'text': 'And righty-left is the area of that rectangle on the side.', 'start': 492.193, 'duration': 3.062}, {'end': 506.701, 'text': 'By the way, I should mention that if you multiply by a constant, say, two times sine of x, things end up a lot simpler.', 'start': 500.219, 'duration': 6.482}, {'end': 515.523, 'text': 'The derivative is just the same as the constant multiplied by the derivative of the function, in this case, two times cosine of x.', 'start': 507.361, 'duration': 8.162}, {'end': 519.544, 'text': "I'll leave it to you to pause and ponder and just kind of verify that that makes sense.", 'start': 515.523, 'duration': 4.021}], 'summary': 'Multiplying by a constant simplifies the derivative, e.g. 2 times sine of x results in 2 times cosine of x.', 'duration': 27.351, 'max_score': 492.193, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA492193.jpg'}], 'start': 260.062, 'title': 'Visualizing functions and composition of derivatives', 'summary': "Explains visualizing the product of two functions as the area of a box using examples, and discusses the influence of a small change in x on the area, ultimately relating it to the derivative. it also discusses the relationship between infinitesimal changes in functions and the derivative, introducing the mnemonic 'left d right, right d left', and demonstrating function composition through examples and visualizations.", 'chapters': [{'end': 395.77, 'start': 260.062, 'title': 'Visualizing functions with area', 'summary': 'Explains how to visualize the product of two functions as the area of a box, using sine of x and x squared as examples, and then discusses the influence of a small change in x on the area, ultimately relating it to the derivative.', 'duration': 135.708, 'highlights': ['The product of two functions can be visualized as the area of a box, with the side lengths representing the functions, such as sine of x and x squared.', 'The value of x can be adjusted, causing changes in the sides of the box, and the area of the box represents the product of the two functions.', 'When considering the influence of a small change in x on the area, the resulting change in area is represented by three snippets of new area, ultimately relating to the derivative.']}, {'end': 592.346, 'start': 396.33, 'title': 'Derivative and function composition', 'summary': "Discusses the derivative of functions, showing the relationship between infinitesimal changes in functions and the derivative, introducing the mnemonic 'left d right, right d left', and demonstrating function composition through examples and visualizations.", 'duration': 196.016, 'highlights': ['The application of derivatives to infinitesimal changes in functions is illustrated, showing the relationship between dx, dx squared, and d sine of x, with dx squared being about 2x times dx and d sine of x being about cosine of x times dx.', 'The derivation of the ratio df divided by dx for any two functions, g and h, is demonstrated as sine of x times the derivative of x squared plus x squared times the derivative of sine.', "The mnemonic 'Left d right, right d left' is introduced to aid in remembering the pattern for calculating derivatives, exemplifying its application with sine of x and x squared, and explaining its representation in the context of an adjustable box.", 'The simplification of derivatives when multiplied by a constant is discussed, exemplifying the derivative of two times sine of x as two times cosine of x.', 'The concept of function composition is introduced, using the example of shoving x squared inside sine of x to obtain the new function sine of x squared, and visualizing the process through three different number lines.']}], 'duration': 332.284, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA260062.jpg', 'highlights': ['The product of two functions can be visualized as the area of a box, with the side lengths representing the functions, such as sine of x and x squared.', 'The value of x can be adjusted, causing changes in the sides of the box, and the area of the box represents the product of the two functions.', 'The application of derivatives to infinitesimal changes in functions is illustrated, showing the relationship between dx, dx squared, and d sine of x, with dx squared being about 2x times dx and d sine of x being about cosine of x times dx.', "The mnemonic 'Left d right, right d left' is introduced to aid in remembering the pattern for calculating derivatives, exemplifying its application with sine of x and x squared, and explaining its representation in the context of an adjustable box.", 'The simplification of derivatives when multiplied by a constant is discussed, exemplifying the derivative of two times sine of x as two times cosine of x.']}, {'end': 934.922, 'segs': [{'end': 761.528, 'src': 'heatmap', 'start': 617.003, 'weight': 0, 'content': [{'end': 626.27, 'text': 'And we could expand this, like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx,', 'start': 617.003, 'duration': 9.267}, {'end': 629.913, 'text': 'but it actually helps to keep things written as dx2, at least for now.', 'start': 626.27, 'duration': 3.643}, {'end': 633.085, 'text': "And in fact, I'm gonna go one step further.", 'start': 631.004, 'duration': 2.081}, {'end': 641.187, 'text': "I'm gonna give a new name to this x squared, maybe h, so that instead of writing dx squared for this nudge, we write dh.", 'start': 633.525, 'duration': 7.662}, {'end': 648.189, 'text': 'And this makes it easier to think about that third value, which is now pegged at sine of h.', 'start': 642.548, 'duration': 5.641}, {'end': 653.631, 'text': 'Its change is d sine of h, the tiny change caused by the nudge dh.', 'start': 648.189, 'duration': 5.442}, {'end': 659.453, 'text': "And, by the way, the fact that it's moving to the left while the dh bump is going to the right.", 'start': 654.871, 'duration': 4.582}, {'end': 664.917, 'text': 'That just means that this change d sine of h is going to be some kind of negative number.', 'start': 659.973, 'duration': 4.944}, {'end': 669.58, 'text': 'And once again we can use our knowledge of the derivative of the sine.', 'start': 666.118, 'duration': 3.462}, {'end': 674.544, 'text': 'This d sine of h is going to be about cosine of h times dh.', 'start': 670.441, 'duration': 4.103}, {'end': 678.026, 'text': "That's what it means for the derivative of sine to be cosine.", 'start': 675.204, 'duration': 2.822}, {'end': 682.63, 'text': 'And unfolding things, we can just replace that h with x squared again.', 'start': 679.267, 'duration': 3.363}, {'end': 689.677, 'text': 'So we know that that bottom nudge is going to have a size of cosine of x squared times dx squared.', 'start': 683.111, 'duration': 6.566}, {'end': 692.399, 'text': "And in fact, let's unfold things even further.", 'start': 690.698, 'duration': 1.701}, {'end': 696.984, 'text': 'That intermediate nudge dx squared is going to be about 2x times dx.', 'start': 692.84, 'duration': 4.144}, {'end': 703.601, 'text': "And it's always a good habit to remind yourself of what an expression like this actually means.", 'start': 699.52, 'duration': 4.081}, {'end': 708.762, 'text': 'In this case, where we started at x equals 1.5 up top.', 'start': 704.341, 'duration': 4.421}, {'end': 720.385, 'text': 'this whole expression is telling us that the size of the nudge on that third line is going to be about cosine of 1.5 squared times 2 times 1.5 times,', 'start': 708.762, 'duration': 11.623}, {'end': 721.906, 'text': 'whatever the size of dx was.', 'start': 720.385, 'duration': 1.521}, {'end': 727.827, 'text': "It's proportional to the size of dx, and this derivative is giving us that proportionality constant.", 'start': 722.646, 'duration': 5.181}, {'end': 732.427, 'text': 'Notice what we came out with here.', 'start': 731.266, 'duration': 1.161}, {'end': 743.091, 'text': "We have the derivative of the outside function and it's still taking in the unaltered inside function and then we're multiplying it by the derivative of that inside function.", 'start': 733.007, 'duration': 10.084}, {'end': 749.246, 'text': 'Again, there is nothing special about sine of x or x squared.', 'start': 745.945, 'duration': 3.301}, {'end': 761.528, 'text': 'If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h,', 'start': 749.726, 'duration': 11.802}], 'summary': 'Derivative calculation using dx and h, involving sine and cosine functions.', 'duration': 95.41, 'max_score': 617.003, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA617003.jpg'}, {'end': 898.552, 'src': 'embed', 'start': 852.262, 'weight': 3, 'content': [{'end': 860.167, 'text': "So notice those dh's cancel out and they give us a ratio between the change in that final output and the change to the input that,", 'start': 852.262, 'duration': 7.905}, {'end': 862.249, 'text': 'through a certain chain of events, brought it about.', 'start': 860.167, 'duration': 2.082}, {'end': 866.94, 'text': 'And that cancellation of dh is not just a notational trick.', 'start': 863.919, 'duration': 3.021}, {'end': 873.843, 'text': "That is a genuine reflection of what's going on with the tiny nudges that underpin everything we do with derivatives.", 'start': 867.28, 'duration': 6.563}, {'end': 883.147, 'text': 'So those are the three basic tools to have in your belt to handle derivatives of functions that combine a lot of smaller things.', 'start': 876.524, 'duration': 6.623}, {'end': 887.189, 'text': "You've got the sum rule, the product rule, and the chain rule.", 'start': 883.848, 'duration': 3.341}, {'end': 890.01, 'text': "And I'll be honest with you,", 'start': 888.57, 'duration': 1.44}, {'end': 898.552, 'text': 'there is a big difference between knowing what the chain rule is and what the product rule is and actually being fluent with applying them in even the most hairy of situations.', 'start': 890.01, 'duration': 8.542}], 'summary': 'Understanding derivatives requires mastering the sum rule, product rule, and chain rule for applying them fluently.', 'duration': 46.29, 'max_score': 852.262, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA852262.jpg'}], 'start': 595.276, 'title': 'Understanding derivatives and the chain rule', 'summary': 'Covers the concept of derivatives using the example of nudging x by dx, and demonstrates how to calculate the derivative of sine and x squared, while also explaining the chain rule for derivatives and the importance of practicing calculus mechanics independently.', 'chapters': [{'end': 732.427, 'start': 595.276, 'title': 'Understanding derivatives through nudging', 'summary': 'Explains the concept of derivatives by using the example of nudging x by dx, and demonstrates how to calculate the derivative of sine and x squared, focusing on the nudge sizes and proportionality constants.', 'duration': 137.151, 'highlights': ['The resulting nudge to the value x squared, the change in x squared caused by dx, is dx squared, and it helps to keep things written as dx squared, at least for now.', 'The change in sine of h caused by the nudge dh is about cosine of h times dh, and we can replace h with x squared, so the bottom nudge is about cosine of x squared times dx squared.', 'The size of the nudge on the third line is about cosine of 1.5 squared times 2 times 1.5 times the size of dx, and this derivative gives us the proportionality constant.']}, {'end': 934.922, 'start': 733.007, 'title': 'Understanding the chain rule', 'summary': 'Explains the chain rule for derivatives, emphasizing the composition of functions and the cancellation of intermediary variables, while highlighting the importance of practicing calculus mechanics independently.', 'duration': 201.915, 'highlights': ['The chain rule for derivatives involves the derivative of the outside function multiplied by the derivative of the inside function, representing the composition of functions.', 'The cancellation of intermediary variables in the chain rule signifies the ratio between the change in the final output and the change in the input that brought it about.', 'Emphasizing the necessity of independent practice for fluency in applying calculus mechanics, including the sum rule, product rule, and chain rule.']}], 'duration': 339.646, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YG15m2VwSjA/pics/YG15m2VwSjA595276.jpg', 'highlights': ['The chain rule for derivatives involves the derivative of the outside function multiplied by the derivative of the inside function, representing the composition of functions.', 'The resulting nudge to the value x squared, the change in x squared caused by dx, is dx squared, and it helps to keep things written as dx squared, at least for now.', 'The change in sine of h caused by the nudge dh is about cosine of h times dh, and we can replace h with x squared, so the bottom nudge is about cosine of x squared times dx squared.', 'Emphasizing the necessity of independent practice for fluency in applying calculus mechanics, including the sum rule, product rule, and chain rule.', 'The size of the nudge on the third line is about cosine of 1.5 squared times 2 times 1.5 times the size of dx, and this derivative gives us the proportionality constant.', 'The cancellation of intermediary variables in the chain rule signifies the ratio between the change in the final output and the change in the input that brought it about.']}], 'highlights': ['The chain rule for derivatives involves the derivative of the outside function multiplied by the derivative of the inside function, representing the composition of functions.', 'The product of two functions can be visualized as the area of a box, with the side lengths representing the functions, such as sine of x and x squared.', 'The chapter emphasizes the importance of having a clear picture in mind for where each derivative comes from', 'The chapter discusses the derivative patterns for products and function composition, which require deeper thinking', 'The chapter explains the three basic ways to combine functions: addition, multiplication, and composition', 'The ratio of the tiny change to the sum function to the tiny change in x is indeed cosine of x plus 2x, the sum of the derivatives of its parts', 'The derivative of sine is cosine, and the change d sine of x is about cosine of x times dx, proportional to the size of the initial nudge dx, and the proportionality constant equals cosine of the input', 'The derivative of x squared is 2x, and the change in the height of the x squared graph is about 2 times x times dx', 'The resulting nudge to the value x squared, the change in x squared caused by dx, is dx squared, and it helps to keep things written as dx squared, at least for now.', 'The application of derivatives to infinitesimal changes in functions is illustrated, showing the relationship between dx, dx squared, and d sine of x, with dx squared being about 2x times dx and d sine of x being about cosine of x times dx.', 'The simplification of derivatives when multiplied by a constant is discussed, exemplifying the derivative of two times sine of x as two times cosine of x.', "The mnemonic 'Left d right, right d left' is introduced to aid in remembering the pattern for calculating derivatives, exemplifying its application with sine of x and x squared, and explaining its representation in the context of an adjustable box.", 'Emphasizing the necessity of independent practice for fluency in applying calculus mechanics, including the sum rule, product rule, and chain rule.', 'The size of the nudge on the third line is about cosine of 1.5 squared times 2 times 1.5 times the size of dx, and this derivative gives us the proportionality constant.', 'The cancellation of intermediary variables in the chain rule signifies the ratio between the change in the final output and the change in the input that brought it about.']}