title

Derivative formulas through geometry | Chapter 3, Essence of calculus

description

Some common derivative formulas explained with geometric intuition.
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Time stamps:
0:00 Intro
1:38 f(x) = x^2
4:41 f(x) = x^3
6:54 f(x) = x^n "Power Rule"
10:07 f(x) = 1/x
12:36 Sine
16:56 Outro
Great video by Think Twice showing this geometric view of the derivative of sin(x):
https://youtu.be/R4o7sraVMZg
Music:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
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detail

{'title': 'Derivative formulas through geometry | Chapter 3, Essence of calculus', 'heatmap': [{'end': 256.416, 'start': 228.642, 'weight': 0.711}, {'end': 722.303, 'start': 647.949, 'weight': 0.734}], 'summary': 'Discusses the importance of computing derivatives in understanding rates of change, explores derivatives geometrically using examples of x^2 and x^3 functions, and delves into the derivative of trigonometric functions such as sine and cosine, providing intuitive explanations and geometric interpretations.', 'chapters': [{'end': 75.465, 'segs': [{'end': 60.258, 'src': 'embed', 'start': 26.806, 'weight': 0, 'content': [{'end': 32.15, 'text': "Maybe it's obvious, but I think it's worth stating explicitly why this is an important thing to be able to do.", 'start': 26.806, 'duration': 5.344}, {'end': 40.997, 'text': "Why much of a calculus student's time ends up going towards grappling with derivatives of abstract functions rather than thinking about concrete rate of change problems?", 'start': 32.63, 'duration': 8.367}, {'end': 49.67, 'text': 'is because a lot of real-world phenomena, the sort of things that we want to use calculus to analyze, are modeled using polynomials,', 'start': 42.284, 'duration': 7.386}, {'end': 53.332, 'text': 'trigonometric functions, exponentials and other pure functions like that.', 'start': 49.67, 'duration': 3.662}, {'end': 60.258, 'text': 'So if you build up some fluency with the ideas of rates of change for those kinds of pure abstract functions,', 'start': 54.133, 'duration': 6.125}], 'summary': 'Understanding abstract functions in calculus is crucial for analyzing real-world phenomena modeled using polynomials, trigonometric functions, and exponentials.', 'duration': 33.452, 'max_score': 26.806, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ26806.jpg'}], 'start': 12.576, 'title': 'Computing derivatives', 'summary': 'Discusses the importance of computing derivatives in understanding rates of change for real-world phenomena modeled using pure abstract functions and concrete situations using calculus.', 'chapters': [{'end': 75.465, 'start': 12.576, 'title': 'Computing derivatives: importance and application', 'summary': 'Discusses the importance of computing derivatives in understanding rates of change for real-world phenomena modeled using pure abstract functions, such as polynomials and trigonometric functions, to analyze concrete situations using calculus.', 'duration': 62.889, 'highlights': ['Real-world phenomena are often modeled using polynomials, trigonometric functions, exponentials, and other pure functions, making it crucial to understand derivatives for these functions.', 'Fluency in the ideas of rates of change for pure abstract functions provides a language to discuss the rates at which things change in concrete situations modeled using calculus.', 'Understanding derivatives is important for analyzing concrete situations using calculus.']}], 'duration': 62.889, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ12576.jpg', 'highlights': ['Real-world phenomena are often modeled using polynomials, trigonometric functions, exponentials, and other pure functions, making it crucial to understand derivatives for these functions.', 'Fluency in the ideas of rates of change for pure abstract functions provides a language to discuss the rates at which things change in concrete situations modeled using calculus.', 'Understanding derivatives is important for analyzing concrete situations using calculus.']}, {'end': 752.952, 'segs': [{'end': 114.666, 'src': 'embed', 'start': 75.465, 'weight': 1, 'content': [{'end': 84.01, 'text': "it's also easy to lose sight of the fact that derivatives are fundamentally about just looking at tiny changes to some quantity and how that relates to a resulting tiny change in another quantity.", 'start': 75.465, 'duration': 8.545}, {'end': 91.374, 'text': 'So in this video and in the next one, my aim is to show you how you can think about a few of these rules intuitively and geometrically.', 'start': 84.69, 'duration': 6.684}, {'end': 96.697, 'text': 'And I really want to encourage you to never forget that tiny nudges are at the heart of derivatives.', 'start': 91.934, 'duration': 4.763}, {'end': 101.262, 'text': "Let's start with a simple function, like f of x equals x squared.", 'start': 97.981, 'duration': 3.281}, {'end': 102.782, 'text': 'What if I asked you its derivative?', 'start': 101.562, 'duration': 1.22}, {'end': 111.185, 'text': 'That is, if you were to look at some value x, like x, equals 2, and compare it to a value slightly bigger, just dx bigger?', 'start': 103.483, 'duration': 7.702}, {'end': 114.666, 'text': "what's the corresponding change in the value of the function df?", 'start': 111.185, 'duration': 3.481}], 'summary': 'Derivatives are about examining tiny changes in quantities and their resulting effects, demonstrated through the example of f(x) = x^2.', 'duration': 39.201, 'max_score': 75.465, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ75465.jpg'}, {'end': 256.416, 'src': 'heatmap', 'start': 228.642, 'weight': 0.711, 'content': [{'end': 233.243, 'text': 'But always remember, in principle, dx should be thought of as a truly tiny amount.', 'start': 228.642, 'duration': 4.601}, {'end': 242.466, 'text': 'And for those truly tiny amounts, a good rule of thumb is that you can ignore anything that includes a dx raised to a power greater than 1.', 'start': 234.064, 'duration': 8.402}, {'end': 245.727, 'text': 'That is, a tiny change squared is a negligible change.', 'start': 242.466, 'duration': 3.261}, {'end': 256.416, 'text': 'What this leaves us with is that df is just some multiple of dx and that multiple 2x, which you could also write as df, divided by dx,', 'start': 247.452, 'duration': 8.964}], 'summary': 'Dx is tiny; ignore dx^2; df is 2 times dx', 'duration': 27.774, 'max_score': 228.642, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ228642.jpg'}, {'end': 447.618, 'src': 'embed', 'start': 414.628, 'weight': 0, 'content': [{'end': 420.153, 'text': "Now, in practice, you wouldn't necessarily think of the square every time you're taking the derivative of x squared,", 'start': 414.628, 'duration': 5.525}, {'end': 424.156, 'text': "nor would you necessarily think of this cube whenever you're taking the derivative of x cubed.", 'start': 420.153, 'duration': 4.003}, {'end': 428.32, 'text': 'Both of them fall under a pretty recognizable pattern for polynomial terms.', 'start': 424.797, 'duration': 3.523}, {'end': 437.147, 'text': 'The derivative of x to the fourth turns out to be 4x cubed, the derivative of x to the fifth is 5x to the fourth, and so on.', 'start': 429.12, 'duration': 8.027}, {'end': 447.618, 'text': "Abstractly, you'd write this as the derivative of x to the n for any power n is n times x to the n minus 1.", 'start': 438.812, 'duration': 8.806}], 'summary': 'Derivative of x^n is n*x^(n-1), e.g., derivative of x^4 is 4x^3.', 'duration': 32.99, 'max_score': 414.628, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ414628.jpg'}, {'end': 722.303, 'src': 'heatmap', 'start': 647.949, 'weight': 0.734, 'content': [{'end': 656.471, 'text': "And let's say that its width is x, which means that the height has to be 1 over x, since the total area of it is 1.", 'start': 647.949, 'duration': 8.522}, {'end': 661.093, 'text': 'So if x was stretched out to 2, then that height is forced down to 1 half.', 'start': 656.471, 'duration': 4.622}, {'end': 665.574, 'text': 'And if you increased x up to 3, then the other side has to be squished down to 1 third.', 'start': 661.833, 'duration': 3.741}, {'end': 670.634, 'text': 'This is a nice way to think about the graph of 1 over x, by the way.', 'start': 667.13, 'duration': 3.504}, {'end': 678.462, 'text': 'If you think of this width x of the puddle as being in the xy plane, then that corresponding output 1 divided by x,', 'start': 671.375, 'duration': 7.087}, {'end': 686.343, 'text': 'the height of the graph above that point is whatever the height of your puddle has to be to maintain an area of 1..', 'start': 678.462, 'duration': 7.881}, {'end': 693.666, 'text': 'So with this visual in mind, for the derivative, imagine nudging up that value of x by some tiny amount, some tiny dx.', 'start': 686.343, 'duration': 7.323}, {'end': 701.189, 'text': 'How must the height of this rectangle change so that the area of the puddle remains constant at 1?', 'start': 694.646, 'duration': 6.543}, {'end': 705.73, 'text': 'That is, increasing the width by dx adds some new area to the right here.', 'start': 701.189, 'duration': 4.541}, {'end': 714.754, 'text': 'So the puddle has to decrease in height by some d1 over x so that the area lost off of that top cancels out the area gained.', 'start': 706.311, 'duration': 8.443}, {'end': 722.303, 'text': "You should think of that d1 over x as being a negative amount, by the way, since it's decreasing the height of the rectangle.", 'start': 716.058, 'duration': 6.245}], 'summary': 'Explains the relationship between width and height of a graph using 1 over x function and derivative.', 'duration': 74.354, 'max_score': 647.949, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ647949.jpg'}], 'start': 75.465, 'title': 'Understanding derivatives geometrically', 'summary': 'Explores derivatives and their geometric interpretations, emphasizing the significance of tiny changes and providing intuitive explanations. it uses examples of x^2 and x^3 functions to illustrate concepts, highlighting the power rule and the relationship between derivatives and geometric reasoning.', 'chapters': [{'end': 160.155, 'start': 75.465, 'title': 'Understanding derivatives intuitively', 'summary': 'Explores the concept of derivatives, emphasizing the importance of tiny changes and demonstrating how to intuitively think about the rules and geometric interpretation. it uses the example of f(x) = x^2 to illustrate the concept of derivatives and the rate of change per unit change in x.', 'duration': 84.69, 'highlights': ['The importance of tiny changes at the heart of derivatives is emphasized, highlighting the concept of looking at tiny changes to quantities and their resulting changes in other quantities.', 'Demonstrates how to intuitively think about the rules and geometric interpretation of derivatives, using the example of f(x) = x^2 to illustrate the concept of derivatives and the rate of change per unit change in x.', 'Explains the concept of finding the derivative of a function, using the example of f(x) = x^2 and analyzing the corresponding change in the value of the function when x is slightly increased.', 'Illustrates the concept of df/dx as the slope of a tangent line to the graph of x^2, showing that the slope generally increases as x increases.', 'Discusses the limitations of understanding the precise formula for a derivative through graphs and emphasizes the need for a more literal look at the meaning of x^2 to comprehend the formula for a derivative.']}, {'end': 366.145, 'start': 160.155, 'title': 'Geometric view of derivatives', 'summary': 'Explains the geometric interpretation of derivatives using the examples of x squared and x cubed functions, demonstrating the concept of tiny changes in area/volume and their relation to dx, with a key highlight being the derivation of the derivative of x squared and x cubed.', 'duration': 205.99, 'highlights': ['The derivative of x squared is 2x, and the derivative of x cubed is 3x squared.', 'The new yellow volume resulting from increasing x by a tiny nudge is mainly comprised of three square faces, totaling to 3x squared dx of volume change.', 'The tiny change in area, df, in the context of x squared, is the result of increasing x by a tiny nudge, dx, and can be represented by two thin rectangles and a minuscule square.']}, {'end': 752.952, 'start': 366.145, 'title': 'Understanding derivatives and the power rule', 'summary': 'Explains the concept of derivatives using the example of x cubed, highlighting the power rule and geometric insights, and moves on to explore the derivative of 1/x with a geometric visualization, emphasizing the importance of understanding derivatives in terms of tiny nudges and geometric reasoning.', 'duration': 386.807, 'highlights': ['The derivative of x to the n for any power n is n times x to the n minus 1, exemplified by the derivative of x cubed being 3x squared.', 'The geometric insight behind derivatives is illustrated with the example of x to the n, emphasizing the negligible contribution of terms beyond n-1 and the importance of understanding derivatives in terms of tiny nudges.', 'The geometric visualization of 1 over x is used to reason about its derivative, emphasizing the concept of maintaining a constant area and the importance of geometric reasoning over blindly applying the power rule.']}], 'duration': 677.487, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ75465.jpg', 'highlights': ['The derivative of x squared is 2x, and the derivative of x cubed is 3x squared.', 'The importance of tiny changes at the heart of derivatives is emphasized, highlighting the concept of looking at tiny changes to quantities and their resulting changes in other quantities.', 'The geometric insight behind derivatives is illustrated with the example of x to the n, emphasizing the negligible contribution of terms beyond n-1 and the importance of understanding derivatives in terms of tiny nudges.', 'The concept of finding the derivative of a function, using the example of f(x) = x^2 and analyzing the corresponding change in the value of the function when x is slightly increased.']}, {'end': 1033.788, 'segs': [{'end': 828.463, 'src': 'embed', 'start': 801.014, 'weight': 2, 'content': [{'end': 807.902, 'text': 'And as your theta value increases and you walk around the circle, your height bobs up and down between negative one and one.', 'start': 801.014, 'duration': 6.888}, {'end': 815.551, 'text': 'So when you graph sine of theta versus theta, you get this wave pattern, the quintessential wave pattern.', 'start': 809.103, 'duration': 6.448}, {'end': 823.08, 'text': 'And just from looking at this graph, we can start to get a feel for the shape of the derivative of the sine.', 'start': 817.737, 'duration': 5.343}, {'end': 828.463, 'text': 'The slope at zero is something positive, since sine of theta is increasing there.', 'start': 823.98, 'duration': 4.483}], 'summary': 'Sine graph shows wave pattern, derivative slope positive at zero.', 'duration': 27.449, 'max_score': 801.014, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ801014.jpg'}, {'end': 874.312, 'src': 'embed', 'start': 847.207, 'weight': 0, 'content': [{'end': 854.191, 'text': "if you're familiar with the graph of trig functions, you might guess that this derivative graph should be exactly cosine of theta,", 'start': 847.207, 'duration': 6.984}, {'end': 859.035, 'text': 'since all the peaks and valleys line up perfectly with where the peaks and valleys for the cosine function should be.', 'start': 854.191, 'duration': 4.844}, {'end': 863.548, 'text': 'And spoiler alert, the derivative is in fact the cosine of theta.', 'start': 860.126, 'duration': 3.422}, {'end': 867.809, 'text': "But aren't you a little curious about why it's precisely cosine of theta?", 'start': 864.268, 'duration': 3.541}, {'end': 874.312, 'text': 'I mean you could have all sorts of functions with peaks and valleys at the same points that have roughly the same shape, but who knows,', 'start': 868.23, 'duration': 6.082}], 'summary': 'The derivative graph of trig functions matches perfectly with cosine of theta.', 'duration': 27.105, 'max_score': 847.207, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ847207.jpg'}, {'end': 1000.133, 'src': 'embed', 'start': 970.464, 'weight': 1, 'content': [{'end': 977.895, 'text': "And from the picture, we can see that that's the ratio between the length of the side adjacent to the angle theta divided by the hypotenuse.", 'start': 970.464, 'duration': 7.431}, {'end': 984.004, 'text': "Well, let's see, adjacent divided by hypotenuse, that's exactly what the cosine of theta means.", 'start': 978.736, 'duration': 5.268}, {'end': 985.867, 'text': "That's the definition of the cosine.", 'start': 984.244, 'duration': 1.623}, {'end': 992.349, 'text': 'So this gives us two different really nice ways of thinking about how the derivative of sine is cosine.', 'start': 987.506, 'duration': 4.843}, {'end': 1000.133, 'text': 'One of them is looking at the graph and getting a loose feel for the shape of things based on thinking about the slope of the sine graph at every single point.', 'start': 993.169, 'duration': 6.964}], 'summary': 'Derivative of sine is cosine, defined as adjacent divided by hypotenuse.', 'duration': 29.669, 'max_score': 970.464, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ970464.jpg'}], 'start': 756.753, 'title': 'Trigonometric functions and derivatives', 'summary': 'Discusses the sine function as a wave pattern, its derivative as the cosine function, and the rationale behind the derivative being precisely cosine of theta. it also explains the derivative of sine as the cosine of theta, using the unit circle and right triangle to illustrate the relationship, providing two different ways of understanding the derivative of sine as cosine, and encourages further exploration to find the derivative of the cosine of theta.', 'chapters': [{'end': 883.889, 'start': 756.753, 'title': 'Trigonometric functions and derivatives', 'summary': 'Discusses the sine function as a wave pattern, its derivative as the cosine function, and the rationale behind the derivative being precisely cosine of theta.', 'duration': 127.136, 'highlights': ['The sine function graph forms a wave pattern with the height of the point above the x-axis bobbing up and down between negative one and one as the theta value increases.', 'The derivative of the sine function is exactly the cosine of theta, as all the peaks and valleys of the derivative line up perfectly with the peaks and valleys of the cosine function.', 'The chapter thoroughly explores the rationale behind the derivative being precisely cosine of theta, addressing the possibility of the derivative being an entirely new type of function with a similar shape.']}, {'end': 1033.788, 'start': 883.889, 'title': 'Understanding derivative of sine and cosine', 'summary': 'Explains the derivative of sine as the cosine of theta, using the unit circle and right triangle to illustrate the relationship, providing two different ways of understanding the derivative of sine as cosine, and encourages further exploration to find the derivative of the cosine of theta.', 'duration': 149.899, 'highlights': ['The derivative of sine is the cosine of theta, which can be understood through the unit circle and right triangle, providing a more precise understanding of the relationship.', 'The chapter encourages further exploration to find the derivative of the cosine of theta, prompting viewers to apply similar reasoning techniques used for understanding the derivative of sine.', 'The video emphasizes understanding derivatives geometrically to make them intuitively reasonable and memorable, aligning with the goal of understanding functions combining simple functions.']}], 'duration': 277.035, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/S0_qX4VJhMQ/pics/S0_qX4VJhMQ756753.jpg', 'highlights': ['The derivative of the sine function is exactly the cosine of theta, aligning perfectly with the cosine function peaks and valleys.', 'The derivative of sine as the cosine of theta can be understood through the unit circle and right triangle, providing a precise understanding of the relationship.', 'The sine function graph forms a wave pattern with the height bobbing between negative one and one as the theta value increases.']}], 'highlights': ['Real-world phenomena are often modeled using polynomials, trigonometric functions, exponentials, and other pure functions, making it crucial to understand derivatives for these functions.', 'Understanding derivatives is important for analyzing concrete situations using calculus.', 'Fluency in the ideas of rates of change for pure abstract functions provides a language to discuss the rates at which things change in concrete situations modeled using calculus.', 'The derivative of x squared is 2x, and the derivative of x cubed is 3x squared.', 'The importance of tiny changes at the heart of derivatives is emphasized, highlighting the concept of looking at tiny changes to quantities and their resulting changes in other quantities.', 'The geometric insight behind derivatives is illustrated with the example of x to the n, emphasizing the negligible contribution of terms beyond n-1 and the importance of understanding derivatives in terms of tiny nudges.', 'The derivative of the sine function is exactly the cosine of theta, aligning perfectly with the cosine function peaks and valleys.', 'The derivative of sine as the cosine of theta can be understood through the unit circle and right triangle, providing a precise understanding of the relationship.', 'The sine function graph forms a wave pattern with the height bobbing between negative one and one as the theta value increases.', 'The concept of finding the derivative of a function, using the example of f(x) = x^2 and analyzing the corresponding change in the value of the function when x is slightly increased.']}