I'm now going to do a bunch more examples
using the chain rule.
So let's see.
Once again.
If I had f of x is equal to, let's see, I don't like this
tool that I'm using now, let's have one of these.
f of x is equal to, say, x to the third plus 2x squared
minus, let's say, minus x to the negative 2.
We haven't put any negative exponents in yet, but I
think you'll see that the same patterns apply.
And all of that to, let's say, the minus seven.
We want to figure out what f prime of x, what the
derivative of f of x is.
So this might seem very complicated and daunting to
you, and obviously to take this entire polynomial to
the negative seventh power would take you forever.
But using the chain rule, we can do it quite quickly.
So the first thing we want to do, is we want to take the
derivative of the inner function, I guess
you could call it.
We want to take the derivative of this.
And what's the derivative of x to the third plus 2x squared
minus x to the negative 2?
Well, we know how to do that.
That was the first type of derivatives we
learned how to do.
It's 3x squared and 2 times 2, plus 4x to the first, or just
4x, and then here, with a negative exponent, we do
the exact same thing.
We say negative 2 times negative 1, right, there's a 1
here, we don't write it down.
So negative 2 times negative 1 is plus 2 x to the, and then we
decrease the exponent by 1, so it's x to the
negative 3, right?
So we figured out what the derivative of the inside is,
and then we just multiply that, that whole thing, times
the derivative of kind of the entire expression.
So then that'll be, we take the minus 7, let me
do a different color.
So this is the entire thing.
So then we take minus 7, so it's times minus 7, this
whole expression, I'm going to run out of space.
x to the third plus 2x squared minus x to the minus 2.
That's minus x to the minus 2.
And all of that, we just decrease this exponent
by 1, to the minus 8.
So let me write it all down a little bit neater now.
So we get f prime of x as the derivative of f of x is equal
to 3x squared plus 4x plus 2x to the minus third power, I
don't know why did that.
That's minus 3.
Times minus seven times x to the third plus 2x squared minus
x to the minus two, all of that to the negative eight power.
And we could simplify it little bit.
Maybe we could just multiply this minus 7, times, we could
distribute it across this expression.
So we'd say, that equals minus 7, so this equals minus 21 x
squared, minus 28 x minus 14x to the negative 3.
All of that times x to the third plus 2 x squared minus x
to the minus 2 to the minus 8.
So there we did it.
We took this, what I would say is a very complicated function,
and using the chain rule and just the basic rules we had
introduced a couple of presentations ago, we were able
to find the derivative of it.
And now, if we wanted to, for whatever application, we could
find the slope of this function at any point x by just
substituting that point into this equation, and we'll get
the slope at that point.
Let me do a slightly harder one, to show you that the chain
rule, you can kind of go arbitrarily deep in
the chain rule.
OK.
So let's say I had, let me see if I can write it
a little bit thinner.
If I had f of x, I don't know if you can see that, I'm going
to do it a little fatter.
f of x is equal to, I want to make it a little bit more
complicated this time.
3x to the minus 2 plus 5 x to the third minus 7x, all of that
to the fifth, and then this whole expression to
the third power.
So I imagine you saying, Sal, you're starting to go nuts,
this is going to take us forever.
Well, I'll show you, using the chain rule, it will
not take that long.
So the way I think about it, so-- f prime of x,
f prime of x equals.
I start off kind with the innermost function.
So let me see if I can use colors to make it
a little bit simpler.
Let's take the derivative of this innermost function first.
Actually, let me give you the big picture.
We want to find the derivative of the innermost function, and
then a little bit bigger, and then a little bit
more big than that.
I know that's not precise mathematical terms, but
you'll get the point when I show you this example.
So first we'll do this inner function, this
inner expression.
And the derivative of that's pretty easy, right?
It's 15x squared minus 7, right?
that was pretty straightforward.
And now we're going to want to multiply that times this
entire derivative here.
So let me circle that in a different-- so then
we want to do this.
We're going to multiply that times this entire derivative.
Well, that's just times 5.
And we just pretend like this is just an x here, right?
Because the derivative of x to the fifth is 5x
to the fourth, right?
But instead of an x, we have this whole expression, 5x
to the third minus 7x.
So we'll write that.
5x to the third minus 7x.
Now the exponent here goes down by one.
So it's 5 times 5x to the third minus 7x, all that
to the fourth power.
So we figured out the derivative of this so far, and
then we want to figure out the derivative of this, so we'll
add it, right, because we're trying to figure the derivative
of this entire expression.
So this is an easy one.
Let me draw that in a different color.
So we want the derivative of this.
So that's negative 2 times 3, so that's negative
6x to the minus three.
So what have we done so far?
We've so far figured out the derivative of this entire
expression, right?
The derivative of that entire expression using
the chain rule is this.
And now, we're almost done.
We just have to multiply that.
So I'm going to just, I've run out of space on that line, but
let's just assume that the line continues.
So that's times.
And now we just take the derivative of kind of
this whole big thing.
And now it's going to be the derivative of, I'm going
to use this brown color.
So it's a whole big expression to the third power, right?
So that becomes times 3 times the whole expression, right?
That's 3 times, now I'm going to write the whole thing, 3x to
the minus 2 plus 5 x the third minus 7x, that to fifth, and
then you decrement this by 1, to the second power.
That was an ultraconfusing example, and this is probably
the hardest chain rule problem you'll see in a lot of
the questions you'll have on your test.
You see, it wasn't that difficult.
We just kind of went to the smallest possible function, and
actually the smallest possible function would have been one of
these terms, but we just found the derivative of this, which
was 15 x squared minus 7, and then we just used the principle
that the derivative of kind of a function is just the
derivative of each of its parts-- well, actually, the
derivative of-- we figured out the derivative of this inner
piece, which was 15x squared minus 7, and then we multiplied
it times the derivative of this slightly larger piece, which is
5 times this entire expression to the fourth, then we added it
to the derivative of 3x to the minus 2.
And then that whole thing, and actually I should put a big
parentheses around here, that whole thing, we multiply it
times the derivative of this larger expression.
I think I might have confused you, so I apologize if I have,
and in the next presentation I'm going to just do a bunch
more chain rule problems, and at some point, it should
start to make sense to you.
I think it's just a matter of seeing example, after
example, after example.
I'll see you into the next presentation, and I apologize
if I have confused you.