[music] Sometimes people ask for the anti derivative.
But instead of talking about the anti derivative it's probably better to talk
about an anti deriviative. So why is this the and an distinction
really important. Because a function can have more than one
anti-derivative. Let's take a look at an example.
Let's think about the function f of x equals 2 x, to find on the whole real
line. Can you think of a function whose
derivative is 2 x? I'm looking for a function which
differentiates to give me this function. So here's an example.
A big f of x equals x squared, and yes, if I differentiate that function I get back
2x, which is in fact little f. So the derivative of big f is little f, or
in other words an anti-derivative for little f.
Is big F. Okay.
Good. Now, can we think of another function
whose derivative is 2x? Well, yeah.
Think about this function big G of x which I'll define to be x squared plus 17 and
big G's derivative is the derivative of this sum which is the derivative of x
squared which is 2x plus the derivative of 17 which is 0.
So big G differentiates to little f. And that means another anti-derivative for
little f is big G. Okay great.
Now let's think of yet another function whose derivative is 2x.
Yeah, I can play this game all day. Big H of x is equal to say x squared plus
123. Big h differentiates to 2 x again.
So big h is another anti derivative of little f.
There's clearly a pattern here, right. What's the general story?
Well here's the general story. So I got this function f of x equals 2 x.
Any anti derivative of little f has the form big f.
Equals x squared plus c for some constant c.
All of out previous examples had this form.
Like here the constant is 0, here the constant is 17, here the constant is 123.
And what I'm claiming here is that any other anti derivative is just x squared
plus some fixed number. How do we know this?
Well, it goes back to the mean value theorem.
So, suppose I've got 2 anti-derivatives, big F and big G for little f.
In other words, I mean that if I differentiate big F I get little f, and if
I differentiate big G I get little f. So, suppose I've got Two anti derivatives
for little F. Now what can I do, like I define a new
function called big H and big H will just be big F minus big G.
Now what do I know about the derivative of big H, the derivative of big H is the
derivative of F minus the derivative of G because the derivative of the difference
is the difference of derivatives. So this is the derivative of f minus the
derivative of g. But what is the derivative of f?
It's little f. What's the derivative of g?
It's little f? This is little f minus little f.
Oh noes. This is 0, right?
The derivative of big H is 0. Now what does that mean?
Well if you think back to the mean value theorem From earlier in the course, if a
functions derivative is equal to zero, and say that function is defined on the whole
real line, like our example with x squared and two x, then, that means that this
function is a constant. Right?
So that means that h of x Is a constant. Now, what that means is that the
difference between F and G, these two antiderivatives for little f, is just some
fixed constant, right? In other words, F of x, is equal to G of x
plus some constant. And this is exactly what we're trying to
get at, right? This idea that if a function's got two
different anti-derivatives. And the function's defined say on the
whole real line. Then those two different anti-derivatives
just differ by a constant or said differently.
If you've got a specific anti-derivative, than any other anti-derivative is just
that specific anti-derivative plus some constant.
Let me say that again, and, and actually write it down.
Suppose that little f is a function that's defined on an interval I.
That's really crucial that it's defined on an interval.
And suppose that big F is an anti-derivative of little f.
In other words if I differentiate big F, I get little f.
Well then any antiderivative of little f, not just this one, any antiderivative has
the form big F of x plus c for some constant c.