- [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.