[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].