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[MUSIC] You're lost. 
You're trapped on a desert island. 

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You have to remember the quotient rule. 
How can you remember the quotient rule? 

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Even while trapped on that desert island, 
you'll remember the vague form of the 

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quotient rule. 
You'll remember, the numerator looks a 

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little bit like the product rule. 
It's a value times a derivative minus the 

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other value times the other derivative. 
But, you might not remember exactly where 

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the minus sign goes, 
right? You don't know if it's 

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f(x)*g(x)-g(x)*f (x), or the other way 
around, g(x)*f(x)-f(x)*g(x). 

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Which one is it? If you weren't trapped 
on the desert island, you'd have access 

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to Wikipedia. 
And you could just look up the quotient 

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rule. 
[MUSIC] Even if you could just look it up 

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let's think about it for a little bit. 
Why is the quotient rule what it is? To 

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make it easy on ourselves let's suppose 
that f of x is positive and g of x is 

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positive. 
Now I'm trying to understand some of the 

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derivative of the quotient which is 
really how is the quotient changing when 

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f and g are doing some changing. 
Let's make that really concrete. 

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Let's suppose that the numerator is 
getting bigger but the denominator is 

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staying the same. 
How does this change? Well then that is 

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bigger, right? If you take Bigger thing 
and cut it into the same number of 

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pieces. 
Then, those pieces are bigger. 

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Now, we could play the game the other 
way. 

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I could keep the pieces the same size, 
but increase the denominator which would 

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be cutting them into more pieces. 
Right? And if I take the same amount of 

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stuff and divide it into more pieces, 
then each of those pieces is smaller. 

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Right? So a same size number divided by a 
bigger number now this fraction is 

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smaller. 
How does this relate to the derivitaive? 

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Well think back to the sign, the s i g n 
of the derivative. 

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So same set up, f of x is positive, g of 
x is positive, maybe the denominator 

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isn't really changing, but the numerator, 
is getting bigger And now I want to know, 

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how is the fraction changing. 
Well, the numerator's getting bigger the 

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denominator's staying the same, the 
fraction should be getting bigger, which 

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then tells us something about the SIGN of 
this derivative. 

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We can similarly analyze the situation 
involving the denominator. 

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So, if the numerator is positive and the 
denominator is positive and the numerator 

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is not really changing, but the 
denominator is getting bigger then the 

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fraction f of x over g of x is getting 
smaller. 

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So that tells us, again, something about 
the sign of the derivative of this ratio. 

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It's negative, because if the 
denominator's getting bigger, and the 

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numerator's not really changing, this 
ratio is getting smaller. 

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How does all of this help us to identify 
the actual quotient rule? How can we get 

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rid of that imposter quotient rule. 
So we've got to guesses as to what the 

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quotient rule might be and I've got some 
information that we just thought about, 

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right if the function's values are 
positive and the numerator's getting 

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bigger and the denominator's not really 
changing that means the fraction's 

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getting bigger. 
If the numerator's not really changing 

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but the denominator's getting bigger then 
that fraction's getting smaller. 

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Now these are truths. 
And which of these truths are compatible 

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with which of these guesses about the 
quotient rule? Well, let's take a look. 

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This first guess about the quotient rule, 
let's see what happens if the numerator's 

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not changing, but the denominator's 
getting bigger. 

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If the numerator's not changing, that 
kills this whole first term and the 

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derivative of f vanishes then. 
But the derivative that a nominator be 

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positive, I'm imagining the value of the 
function is positive. 

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So, this is a positive number but a 
negative sign there, so this is now a 

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negative numerator divided by a positive 
number. 

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So, if this were the quotient rule, it 
would be telling us that an increasing 

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denominator makes this ratio smaller. 
It makes the derivative negative, 

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that's good. 
That's really compatible with this 

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picture. 
Now, is this compatible with that? Well, 

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what would happen here if the denominator 
were increasing, but the numerator was 

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staying the same? If the numerator's 
staying the same, this is zero, which 

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kills this term. 
And I'm just left with this, 

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and if the numerator is increasing then 
this term is positive and imagine the 

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function's [UNKNOWN] positive. 
So you got a positive thing divided by 

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positive thing. 
If these were the quotient row, an 

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increasing denominator where the 
numerator remains the same would make 

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this ratio Increase because this 
derivative would be positive or this 

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can't be, this isn't the quotient rule, 
it's not compatible with this fact. 

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In fact, this is the quotient rule and we 
can see that it's also compatible with 

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this first fact. 
If the numerator is getting bigger but 

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the denominator is staying the same well 
the denominator staying the same makes 

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this term zero which kills this whole 
term and all I'm left with is this. 

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Now imagine g is positive and the 
derivative of the numerator is positive. 

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The derivator is getting bigger. 
This is positive, 

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I've got a positive thing divided by a 
positive thing. 

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That makes the derivative positive. 
And that makes sense. 

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If the numerator's getting bigger and the 
denominator's staying the same, the 

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derivative is positive and that's exactly 
what this true quotient rule is saying. 

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Fundamentally, I don't want you just to 
memorize all of these rules. 

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[MUSIC] I want you to understand why the 
rules are what they are. 

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I want you to get a feeling for why 
there's a negative sign in the quotient 

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rule. 
It really belongs there. 

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[MUSIC].