[music] Differentiating is a process really guaranteed to work.
Alright, if somebody writes down a function and asks you to differentiate it.
You can. It might be painful, alright, it might
involve a lot of steps but it's just a matter of applying those steps carefully.
In contrast antidifferentiation might be really hard.
Let's try a really hard example now. For instance, can you find some function
so if I differentiate that function, I get e to the negative x squared?
Which is just another way of saying, I want to anti-differentiate e to the
negative x squared. Right, what is that?
Well, let's make a guess and see if our guess is correct.
I mean maybe, an antiderivative of e to the negative x squared is I don't know, e
to the negative x squared, over minus 2 x plus c, right.
I mean I'm not saying this is correct, but it could be true.
If this were the case, I can check it, right.
I can differentiate this side and see if I get this.
So let's differentiate e to the minus x squared over minus 2 x.
That's sort of the same thing as differentiating e to the minus x squared
times minus 2 x to the negative 1st power. So that's a derivative of a product, which
is assuming the product rule's good for, right?
So first I'll differentiate e to the minus x squared.
And I'll multiply that by the second term minus two x to the minus 1st power and
I'll add to that the first term, e to the minus x squared times the derivative of
the second term, which is minus two x to the minus 1st power.
But what's the derivative of e to the minus x squared?
That's e to the minus x square cause the derivative of e to the is e to the.
Times the derivative of minus x squared which is minus 2 x.
And then it's times negative 2 x to the negative 1st power, right?
Plus e to the minus x squared times the derivative of minus 2 x to the minus 1st
power. I'll just copy that down and we'll deal
with it in a second. Because this is pretty exciting right here
right. This term cancels, this term cancels, this
is looking pretty good right. Maybe the antiderivative of e to the minus
x squared really is this. Because when I differentiate this thing,
at least I get a e to the minus x squared. The bad news is that I also get this term,
right. And this terms out to be e to the minus x
squared, now I gotta deal with this. It's again e to the minus x squared and
then times the derivative of this. Well, when I differentiate this, I get a
negative sign times this thing to the negative 2nd power, right?
And then by the chain rule, the derivative of the inside, which is negative 2.
And yeah, I mean this is, this is bad right.
I mean I differentiated this thing right here and I got back some of that included
in the e to the minus x squared but also includes non zero term.
So as result this is not the case. Well that didn't work but it wasn't from
lack trying. In fact, an antiderivative for e to the
negative x squared cannot be expressed using elementary functions.
What do I mean by elementary functions? I mean things like polynomials, trig
functions, e to the x, log, things like that, right?
I mean, this is really a surprising result.
I mean there is a function whose derivative is e to the negative x squared,
but it's not a function I can write down. Using the functions that I already have in
hand. May be e to the negative x squared just an
isolated terrible example. I can't answer differentiate e to the e to
the x using elementary functions. I can't answer differentiate log, log x
using elementary functions. I can't even answer differentiate this
very reasonable looking algebraic function.
The square root of 1 plus x to the 4th. I can't find an anti-derivative of this
just using elementary functions. So, what does this mean?
Let's try to summarize the situation. It's not just that many functions are hard
to antidifferentiate. It's that many functions are impossible to
antidifferentiate, not in the sense that they don't have an antiderivative.
But in the sense that we're not going to succeed in writing down that
antiderivative using the functions that we have at hand.
In light of the difficulties of antidifferentiating, the fact that there's
no guaranteed answer, it means that we should be happy that we can ever evaluate
an antiderivative problem right. It also reflects the fact that some real
creativity is needed. You know, if an answer's not guaranteed
then potentially we're going to require more creativity to cook up the answers,
even when they exist.