[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]