[MUSIC]. 
You've heard of the, ask not what your 
country can do for you, but what you can 
do for your country rule. 
These chiastic rules for limits. 
The limit's the sum of the limits. 
Same thing is true for derivatives. 
Let's go to the board. 
Here's the rule for derivatives. 
The derivative of f + g. 
Is the derivative of f + the derivative 
of g. 
In short, the derivative of the sum is 
the sum of the derivatives. 
Why does this make sense? Well, think 
back to what the derivative is measuring. 
The derivative is measuring how changing 
the input affects the output. 
In this case, I want to know how changing 
x affects the sum of f of x and g of x. 
Well, the sum is affected by the sum of 
the effects. 
The sum of the derivative of f and the 
derivative of g. 
Let' see this in a specific case. 
Here's a specific case. 
The function f(x) = x^3 + x^2. 
Let's differentiate this. 
I'm going to calculate d / dx 
(x^3+3=x^2). 
Now this is a derivative of a sum, which 
is the sum of derivatives. 
Now we have to figure out what's the 
derivative of x ^ 3 and what's the 
derivative of x^2. 
That's the power law. 
derivative of x^3 is 3x^2, 
and the derivative of x^2 is 2x. 
And now there's no more d/dx's. 
We've calculated the derivative. 
The derivative of x^3+x^2+2x. 
Once we know this, we can figure out 
where the derivative is positive and 
where it's negative. 
So, the derivative was 3x^3+2x and I want 
to know where that's positive and 
negative, which values of x make that 
bigger than 0, which values of x make 
that less than 0. 
one approach to thinking about this is to 
factor through the x^2+2x. 
I can write that as x(3x+2). 
And once I factor it like this I can 
figure out the SIGN of this by figuring 
out the SIGN of these two terms 
separately. 
visualize this as a direct whole number 
line. 
So x, here's a number line. 
X is positive when it is bigger than 0 
and negative when X is less than 0. 
That's not too complicated. 
Well look 3x+2. 
Well, 3x+2, I draw a number line for 
that. 
The exciting point is -2/3. 
When x is less than -2/3, 3x+2 is 
negative. 
And when x is bigger than -2/3's, then 
3x+2. 
Is positive. 
Now, I really don't care about x and 3x+2 
separately. 
I want to put them together right. 
I want to know when their product is 
positive or negative. 
So, I write down the product x * 3x+2 
make a new number line here. 
I'll record both of these points -2/3 and 
0 then I can think about what happens. 
When x is less than -2/3 then x is 
negative and 3x+2 is negative. 
So the product is a negative * a 
negative, which is positive. 
When x is between -2/3 and 0, then x is 
negative but 3x+2 is positive and a 
negative times a positive number is 
negative. 
And finally, when x is bigger than 0, 
well then x is positive and also 3x+2 is 
positive. 
So the Is positive. 
So, here on this number line, I've 
recorded the information about when 3x^2 
+ 2x is positive or negative. 
Now we can use this information to say 
something about the graph. 
Here's the graph of the function x^3+x^2. 
Goes up, down and up, And that's exactly 
what you'd expect from the derivative, 
right? We calculated before that if 
you're standing to the left of - 2/3's, 
then the derivative was positive, and 
indeed the functions going up. 
Up. 
Now once you get to -2/3, the derivative 
is 0 but then over here, between -2/3 and 
0 the derivative is negative and indeed 
the graph is moving down until you get to 
0 when the derivative of this function is 
positive again and the graph is going up. 
Look, the sign of the derivative 
Positive, negative, positive is reflected 
in the direction that this graph is 
moving. 
Increasing, decreasing, increasing. 
Incredible. 
By being able to differentiate x^2+x^3, 
we're able to gain real insight into the 
graph of the function. 
We're not just plotting a whole bunch of 
points and hoping that we can fill it in 
with a straight line. 
By looking at the derivative, we know 
that the function is increasing and 
decreasing. 
We're able to say something, for sure. 
[MUSIC]