Reverse chain rule
Reverse chain rule introduction
- [Voiceover] Hopefully we all remember our good friend the chain rule from differential calculus that tells us that if I were to take the derivative with respect to x of g of f of x, g of, let me write those parentheses a little bit closer, g of f of x, g of f of x, that this is just going to be equal to the derivative of g with respect to f of x, so we can write that as g prime of f of x. G prime of f of x, times the derivative of f with respect to x, times f prime of x. And if you want to see it in the other notation, I guess you could say, it would be, you could write this part right over here as the derivative of g with respect to f times the derivative of f. The derivative of f with respect to x, and that's going to give you the derivative of g with respect to x. This is just a review, this is the chain rule that you remember from, or hopefully remember, from differential calculus. It's hard to get, it's hard to get too far in calculus without really grokking, really understanding the chain rule. So what I want to do here is, well if this is true, then can't we go the other way around? If I wanted to take the integral of this, if I wanted to take the integral of g prime of f of x, g prime of f of x, times f prime of x, dx, well, this should just be equal to, this should just be equal to g of f of x, g of f of x, and then of course whenever I'm taking an indefinite integral g of, let me make sure they're the same color, g of f of x, so I just swapped sides, I'm going the other way. So if I'm taking the indefinite integral, wouldn't it just be equal to this? And of course I can't forget that I could have a constant here now that might have been introduced, because if I take the derivative, the constant disappears. And so this idea, you could really just call the reverse chain rule. Reverse, reverse chain, the reverse chain rule. Which is essentially, or it's exactly what we did with u-substitution, we just did it a little bit more methodically with u-substitution. And we'll see that in a second, but before we see how u-substitution relates to what I just wrote down here, let's actually apply it and see where it's useful. And this is really a way of doing u-substitution without having to do u-substitution, or doing u-substitution in your head, or doing u-substitution-like problems a little bit faster. So let me give you an example. So let's say that we had, and I'm going to color code it so that it jumps out at you a little bit more, let's say that we had sine of x, and I'm going to write it this way, I could write it, so let's say sine of x, sine of x squared, and obviously the typical convention, the typical, the sine of x squared, the typical convention would be to put the squared right over here, but I'm going to write it like this, and I think you might be able to guess why. Sine of x squared times cosine of x. Times, actually, I'll do this in a, let me do this in a different color. Times cosine of x, times cosine of x. So I encourage you to pause this video and think about, does it meet this pattern here, and if so, what is this indefinite integral going to be? Well let's think about it. If f of x is sine of x, what's the derivative of that? What's f prime of x? Well f prime of x in that circumstance is going to be cosine of x, and what is g? Well g is whatever you input into g squared. So what's this going to be if we just do the reverse chain rule? Well this is going to be, well we take sorry, g prime is taking whatever this thing is, squared, so g is going to be the anti-derivative of that, so it's going to be taking something to the third power and then dividing it by three, so let's do that. So if we essentially take the anti-derivative here with respect to sine of x, instead of with respect to x, you're going to get you're going to get sine of x, sine of x to the, to the third power over three, and then of course you have the, you have the plus c. And if you don't believe this, just take the derivative of this, you'll have to employ the chain rule and you'll get exactly this. And you say well wait, how does this relate to u-substitution? Well in u-substitution you would have said u equals sine of x, then du would have been cosine of x, dx, and actually let me just do that. That actually might clear things up a little bit. You would set this to be u, and then this, all of this business right over here, would then be du, and then you would have the integral, you would have the integral u squared, u squared, I don't have to put parentheses around it, u squared, du. U squared, du, well, let me do that in that orange color, u squared, du. Well that's pretty straightforward, this is going to be equal to u, this is going to be equal to u to the third power over three, plus c, which is equal to what? Well we just said u is equal to sine of x, you reverse substitute, and you're going to get exactly that right over here. So when we talk about the reverse chain rule, it's essentially just doing u-substitution in our head. So in the next few examples, I will do exactly that.