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[music] Differentiating is a process really guaranteed to work.

Â Alright, if somebody writes down a function and asks you to differentiate it.

Â You can. It might be painful, alright, it might

Â involve a lot of steps but it's just a matter of applying those steps carefully.

Â In contrast antidifferentiation might be really hard.

Â Let's try a really hard example now. For instance, can you find some function

Â so if I differentiate that function, I get e to the negative x squared?

Â Which is just another way of saying, I want to anti-differentiate e to the

Â negative x squared. Right, what is that?

Â Well, let's make a guess and see if our guess is correct.

Â I mean maybe, an antiderivative of e to the negative x squared is I don't know, e

Â to the negative x squared, over minus 2 x plus c, right.

Â I mean I'm not saying this is correct, but it could be true.

Â If this were the case, I can check it, right.

Â I can differentiate this side and see if I get this.

Â So let's differentiate e to the minus x squared over minus 2 x.

Â That's sort of the same thing as differentiating e to the minus x squared

Â times minus 2 x to the negative 1st power. So that's a derivative of a product, which

Â is assuming the product rule's good for, right?

Â So first I'll differentiate e to the minus x squared.

Â And I'll multiply that by the second term minus two x to the minus 1st power and

Â I'll add to that the first term, e to the minus x squared times the derivative of

Â the second term, which is minus two x to the minus 1st power.

Â But what's the derivative of e to the minus x squared?

Â That's e to the minus x square cause the derivative of e to the is e to the.

Â Times the derivative of minus x squared which is minus 2 x.

Â And then it's times negative 2 x to the negative 1st power, right?

Â Plus e to the minus x squared times the derivative of minus 2 x to the minus 1st

Â power. I'll just copy that down and we'll deal

Â with it in a second. Because this is pretty exciting right here

Â right. This term cancels, this term cancels, this

Â is looking pretty good right. Maybe the antiderivative of e to the minus

Â x squared really is this. Because when I differentiate this thing,

Â at least I get a e to the minus x squared. The bad news is that I also get this term,

Â right. And this terms out to be e to the minus x

Â squared, now I gotta deal with this. It's again e to the minus x squared and

Â then times the derivative of this. Well, when I differentiate this, I get a

Â negative sign times this thing to the negative 2nd power, right?

Â And then by the chain rule, the derivative of the inside, which is negative 2.

Â And yeah, I mean this is, this is bad right.

Â I mean I differentiated this thing right here and I got back some of that included

Â in the e to the minus x squared but also includes non zero term.

Â So as result this is not the case. Well that didn't work but it wasn't from

Â lack trying. In fact, an antiderivative for e to the

Â negative x squared cannot be expressed using elementary functions.

Â What do I mean by elementary functions? I mean things like polynomials, trig

Â functions, e to the x, log, things like that, right?

Â I mean, this is really a surprising result.

Â I mean there is a function whose derivative is e to the negative x squared,

Â but it's not a function I can write down. Using the functions that I already have in

Â hand. May be e to the negative x squared just an

Â isolated terrible example. I can't answer differentiate e to the e to

Â the x using elementary functions. I can't answer differentiate log, log x

Â using elementary functions. I can't even answer differentiate this

Â very reasonable looking algebraic function.

Â The square root of 1 plus x to the 4th. I can't find an anti-derivative of this

Â just using elementary functions. So, what does this mean?

Â Let's try to summarize the situation. It's not just that many functions are hard

Â to antidifferentiate. It's that many functions are impossible to

Â antidifferentiate, not in the sense that they don't have an antiderivative.

Â But in the sense that we're not going to succeed in writing down that

Â antiderivative using the functions that we have at hand.

Â In light of the difficulties of antidifferentiating, the fact that there's

Â no guaranteed answer, it means that we should be happy that we can ever evaluate

Â an antiderivative problem right. It also reflects the fact that some real

Â creativity is needed. You know, if an answer's not guaranteed

Â then potentially we're going to require more creativity to cook up the answers,

Â even when they exist.

Â