[MUSIC] You're lost. You're trapped on a desert island.
You have to remember the quotient rule. How can you remember the quotient rule?
Even while trapped on that desert island, you'll remember the vague form of the
quotient rule. You'll remember, the numerator looks a
little bit like the product rule. It's a value times a derivative minus the
other value times the other derivative. But, you might not remember exactly where
the minus sign goes, right? You don't know if it's
f(x)*g(x)-g(x)*f (x), or the other way around, g(x)*f(x)-f(x)*g(x).
Which one is it? If you weren't trapped on the desert island, you'd have access
to Wikipedia. And you could just look up the quotient
rule. [MUSIC] Even if you could just look it up
let's think about it for a little bit. Why is the quotient rule what it is? To
make it easy on ourselves let's suppose that f of x is positive and g of x is
positive. Now I'm trying to understand some of the
derivative of the quotient which is really how is the quotient changing when
f and g are doing some changing. Let's make that really concrete.
Let's suppose that the numerator is getting bigger but the denominator is
staying the same. How does this change? Well then that is
bigger, right? If you take Bigger thing and cut it into the same number of
pieces. Then, those pieces are bigger.
Now, we could play the game the other way.
I could keep the pieces the same size, but increase the denominator which would
be cutting them into more pieces. Right? And if I take the same amount of
stuff and divide it into more pieces, then each of those pieces is smaller.
Right? So a same size number divided by a bigger number now this fraction is
smaller. How does this relate to the derivitaive?
Well think back to the sign, the s i g n of the derivative.
So same set up, f of x is positive, g of x is positive, maybe the denominator
isn't really changing, but the numerator, is getting bigger And now I want to know,
how is the fraction changing. Well, the numerator's getting bigger the
denominator's staying the same, the fraction should be getting bigger, which
then tells us something about the SIGN of this derivative.
We can similarly analyze the situation involving the denominator.
So, if the numerator is positive and the denominator is positive and the numerator
is not really changing, but the denominator is getting bigger then the
fraction f of x over g of x is getting smaller.
So that tells us, again, something about the sign of the derivative of this ratio.
It's negative, because if the denominator's getting bigger, and the
numerator's not really changing, this ratio is getting smaller.
How does all of this help us to identify the actual quotient rule? How can we get
rid of that imposter quotient rule. So we've got to guesses as to what the
quotient rule might be and I've got some information that we just thought about,
right if the function's values are positive and the numerator's getting
bigger and the denominator's not really changing that means the fraction's
getting bigger. If the numerator's not really changing
but the denominator's getting bigger then that fraction's getting smaller.
Now these are truths. And which of these truths are compatible
with which of these guesses about the quotient rule? Well, let's take a look.
This first guess about the quotient rule, let's see what happens if the numerator's
not changing, but the denominator's getting bigger.
If the numerator's not changing, that kills this whole first term and the
derivative of f vanishes then. But the derivative that a nominator be
positive, I'm imagining the value of the function is positive.
So, this is a positive number but a negative sign there, so this is now a
negative numerator divided by a positive number.
So, if this were the quotient rule, it would be telling us that an increasing
denominator makes this ratio smaller. It makes the derivative negative,
that's good. That's really compatible with this
picture. Now, is this compatible with that? Well,
what would happen here if the denominator were increasing, but the numerator was
staying the same? If the numerator's staying the same, this is zero, which
kills this term. And I'm just left with this,
and if the numerator is increasing then this term is positive and imagine the
function's [UNKNOWN] positive. So you got a positive thing divided by
positive thing. If these were the quotient row, an
increasing denominator where the numerator remains the same would make
this ratio Increase because this derivative would be positive or this
can't be, this isn't the quotient rule, it's not compatible with this fact.
In fact, this is the quotient rule and we can see that it's also compatible with
this first fact. If the numerator is getting bigger but
the denominator is staying the same well the denominator staying the same makes
this term zero which kills this whole term and all I'm left with is this.
Now imagine g is positive and the derivative of the numerator is positive.
The derivator is getting bigger. This is positive,
I've got a positive thing divided by a positive thing.
That makes the derivative positive. And that makes sense.
If the numerator's getting bigger and the denominator's staying the same, the
derivative is positive and that's exactly what this true quotient rule is saying.
Fundamentally, I don't want you just to memorize all of these rules.
[MUSIC] I want you to understand why the rules are what they are.
I want you to get a feeling for why there's a negative sign in the quotient
rule. It really belongs there.
[MUSIC].