Interpreting the Chain Rule Graphically
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- Опубліковано 27 вер 2024
- Description:
We try to visualize what is going on with the chain rule graphically.
Learning Objectives:
1) Find the slope of a a composite function described graphically.
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!
Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.
This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.
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Amazing! I never saw an explanation with "Look at where it would go with the standard derivative of f(x) at X, and then multiply to sort of fix the value to what it really is". This was enlightening.
Thank you!
This pairs well with the idea that 3Blue1Brown has put forward, that derivative tells us how a function stretches space.
You said this interpretation generalizes, but you didn't really show how, so let me do that.
We can interpret f(g(x)) like that: g(x) takes the x axis, and transforms it - it puts the markings into its own order (for instance, it could make marking 1 go between markings 2 and 3, and make the distance between 1 and 3 equal to the distance between 1 and 2. Remember, this occurs with EVERY marking, that is, every number. Not just whole or natural numbers, all the real numbers (as long as g(x) is even defined on them) are put into some kind of wicked order). After transforming it, it plugs that x axis into f(x), who gives back its own graphic based on the wicked x-axis. But we have a messed up x-axis now - so we undo the transformation of g(x) on every x coordinate. Voila - now we have the graphic of f(g(x)).
If we look at a very small portion of the x-axis, though, g(x) doesn't just transform it arbitrarily - it stretches and moves it around, in fact! At small enough scales, we can view g(x) as just a linear function. In fact, the factor, by which g(x) stretches the tiny portion of x-axis around the point x1, is equal to g'(x1)!
With that in mind, let's look at f(g(x)) in the tiny part of x-axis around x1, and keep track of the derivative of the function.
g(x) takes that portion, and linearly transforms it. It stretches it by k, where k = g'(x1). Then, f(x) gives its own graph at the now stretched and moved portion of the x-axis. Finally, we actually have a graph - so we can now find the derivative. So, right now the derivative (which is the slope of the tangent line, remember) is equal to f'(g(x1)), obviously. However! Now, we apply the inverse of the linear transformation g(x) gives us around x1. We divide every x coordinate on the graph by k and move it around. But what happens to the slope of the tangent line - that is, what happens to the derivative? We know that every x is divided by k, so the slope of the line must be multiplied by it! Therefore, now, at the final step, the slope of the tangent line at x1 of f(g(x)) = (f(g(x)))' = f'(g(x1)) * k = f'(g(x1)) * g'(x1).
I request you to make a video which depicts graphical meaning of successive differentiation.
I finished calc 4 in the winter and I always thought of the derivative of a function as a different function with unique characteristics.
Although obvious now, I never thought to consider it as a transformation of the original or its components. Interesting.
In a manner of speaking, can we look at derivatives as a type of transformation, and is there any insight gained from this viewpoint that immediately comes to mind?
Hi, Which software do you use to make your video?
It would be nice if you show how to obtain the chain rule from the definition of the derivative
Could be better... started great 👍... ended confusing... it's like you was explaining where anyone could understand i.e. ... a layman... then you turned into a peacock 🦚 and flex'd your intelligence past those lose keen... 🍺 One beer out of 5 ... swill rating
great video - thanks!