![]() And that became our new line’s slope.Ĭhain rule! How we generalize this to the chain ruleįor any composition of functions, we are going to have an inner and an outer function. Where did we ultimately get the slope of 2 from? When we composed to two lines together, we multiplied the slope of the inner function (12) by the slope of the outer function (1/6). Let’s look at a graph of and our tangent line: We are dealing with which really means that we’re taking the square root of 9. ![]() Now let’s start with the square root function. At, the tangent line is (I’m not showing the work, but you can trust me that it’s true, or work it out yourself.) I’m going to argue that just as and are composed to get our final function, we can compose the tangent lines to these two functions to get the final tangent line at. Let’s see if I can’t clear this up by making it concrete with an example.Īnd so we can be super concrete, let’s try to find, which is simply the slope of the tangent line of at. We can look at a composition of functions at a point as simply a composition of these little line segments. So why not look at function composition in the same way? My Thought For Using This for The Chain Rule The whole point of this is to show that tangent lines undergo the same transformations as the functions - because the functions themselves are pretty much just a bunch of these infinitely tiny tangent line segments all connected together! So it would actually be weird if the tangent lines didn’t behave like the functions. Yay! It worked! (But of course we knew that would happen.) Now let’s plot and our transmogrified tangent line: Now let’s put that tangent line through the transformations: We see that is secretly which has undergone a vertical stretch of 2, a horizontal shrink of 1/5, and has been moved up 1. ![]() In my class, we’ve learned that whatever transformations a function undergoes, the tangent line undergoes the same transformations! If you want to see that, you can check it out here. Thus, when we take a derivative, we’re pretty much just asking “what’s the slope of the little line segment at ?” for example. Every “nice” function (and those are the functions we’re dealing with) is basically like an infinite number of little line segments connected together. I am not yet sure if I have a way to turn this into something that my kids will understand. It’s for me to work through some unformed ideas. I only had the insight 10 minutes ago so I’m going to use this blogpost to see if I can’t get the ideas straight in my head… The point of this post is not to share a way I’ve made the chain rule understandable. I think I now have a way that might help students to get conceptually understand what’s going on. (The gear thing doesn’t help me get it… Although I understand the analogy, it feels divorced from the actual functions themselves… and these functions have a constant rate of change.) But I have never yet found a way to conceptually get them to understand it without confusing them. I have found ways to help kids remember the chain rule (“the outer function is the mama, the inner function is the baby… when you take the derivative, you derive the mama and leave the baby inside, and then you multiply by the derivative of baby”), ways to write things down so their information stays organized, and I have shown them enough patterns to let them see it’s true. I’m soon going to embark on teaching the chain rule in calculus.
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