We've seen what happens when we 
differentiate the accumulation function. 
That's the fundamental theorem of 
calculus. 
If I take the derivative of the integral 
from a to b of f of x dx, well what do I 
get? 
Well, it's just f. 
Of b, right? 
The rate of the change of the 
accumulation function is the functions 
value. 
But, what would happen if I were to 
differentiate with respect to the left 
hand end point instead of with respect to 
the right hand end point? 
By which I mean. 
What's the derivative with respect to a 
of the integral from a to b of f of x dx. 
Right. 
What happens if instead of diferentiating 
with respect to the top end point, I'm 
diferentiating with respect to the bottom 
end point? 
I'm asking how is this function as a 
function of a changing? 
Let's think about the graph. 
Let's draw my coordinate axes. 
And I'll draw some random looking 
function. 
And I'll pick some points a and b. 
And the integral from a to b calculates 
the area in here. 
And I want to know how does that integral 
change when I wiggle a? 
I'm asking to differentiate the integral 
with respect to this. 
Left-hand endpoint. 
So let's wiggle the left-hand endpoint. 
Let's move it over a little bit to a plus 
h. 
I'm imagining h is very small. 
Alright, and I want to know, how does the 
integral change? 
Well the quantity that calculates the 
absolute change in the integral is this. 
What's this thing here? 
This is the integral from a plus h to b. 
And this thing here is the integral from 
a to b, so this difference is telling me 
how the integral changes when I replace a 
by a plus h. 
Now if you think about it, what this is 
really calculating is, you know, related 
to the integral from just a plus h, 
right. 
This integral is calculating this area, 
and subtracting this larger area. 
So the difference is really just this 
area in here between A and A+H but it 
comes with a negative sign because I'm 
subtracting this smaller area and I'm 
subtracting now this larger area. 
So I've got this negative sign here. 
Now this region, if h is small enough, is 
practically a rectangle, and it's 
practically a rectangle of width h and 
height let's say f of a. 
So, that means that this difference is at 
least approximately just h times the 
function's value at a. 
Now how does that help? 
Remember what I'm trying to calculate. 
I, I'm trying to differentiate the 
interval from a to be with respect to a. 
That means I'm trying to take the limit 
of this difference quotient. 
But the numerator here, we just saw, is 
approximately, negative h times f of a. 
And that means in the limit I expect to 
get an answer of just negative f of a, 
alright, these hs will cancel in the 
limit, and that's exactly what I hope 
for, right? 
The derivitive of the integral from a to 
b with respect to the left hand endpoint 
a is negative f of a. 
So I can summarize this. 
So I can summarize this. 
The derivative with respect to a of the 
integral from a to b of f of x dx is 
negative f of a. 
This fact coheres with a certain 
convention about integration. 
The convention that we use is that if we 
integrate from a to b. 
the function f of x dx that's negative 
the interval from b to a of f of x dx. 
So if you integrate the wrong way, so to 
speak, we want to count that as negative 
area. 
 >> Now in light of this convention, what do 
we know? 
 >> So d da of the integral from a to b of f 
of x dx is d da of this, negative the 
integral from b to a of f of x dx. 
But now this is the derivative of the top 
end point which is just the usual 
fundamental theorem of calculus. 
So this is negative f of a. 
So the upshot here is that 
differentiating the integral from a to b, 
with respect to the left hand or bottom 
end point, Is negative the function's 
value. 
And that make sense because if you 
increase the left hand endpoint that 
decreases the area. 
Alright, so it seems reasonable that a 
negative sign should be popping up there. 
And the cool thing is that, that fact is 
cohering with a convention about 
integration. 
That compared to integrating the usual 
way, if you integrate the wrong way you 
introduce a negative sign.