CHAPTER 01.07: TAYLOR SERIES REVISITED: Taylor Series: Deriving the Series for exp(x)      In this segment, we're going to take an example of Taylor series. And we're going to try to find out the Maclaurin series for e to the power x. So we already know that the Maclaurin series, which you must have seen in handbooks and other calculus classes, looks like this. So let's go ahead and find out how can we find out this?  So we'll derive this, as opposed to taking this as something which is given to you, we're going to derive this particular formula for e to the power x, this series formulation for e to the power x, from Taylor's . . . Taylor series.  So if you look at Taylor's Theorem, we have that the value of the function at x plus h is given by that, if I know the value of the function at x, plus I know the value of the first derivative of the function at x, plus I know the second derivative of the function at x, plus I know the third derivative of the function at x, and all the other derivatives.  So it's not just a question of knowing just the first few derivatives, but also knowing the function itself at that particular point, the derivative of the function at that particular point, the second derivative, third derivative, and so on and so forth, what you can do is you can find the value of the function at some other point.  So let's go ahead and use this to be able to derive . . . derive this.  So what I'm going to do is, in order to be able to do that, I do need to know what the derivatives are.    So my function f of x is e to the power x, let's suppose.  Now what is f prime of x, from your differential calculation . . . calculus class?  It is e to the power x also, right?  And we also know the second derivative is e to the power x, the third derivative is also e to the power x, okay?  Pretty straightforward, because we know that the derivative of e to the power x is just e to the power x itself, so all the derivatives are of the form e to the power x. Now, what we're going to do is, one of the things which we do know about the . . . about these values of e to the power x is that we know the value at x equal to 0, e to the power 0 is 1.  So we don't know what the . . . we don't directly know what the value of e to the power something is, other than that of 0.  Of course, we know that as x approaches minus infinity, it turns out to be 0, and when x equal to 0, we know it's exactly equal to 1.  So we know that the value of the function at 0 is 1, the value of the derivative of the function at 0 is 1, the value of the second derivative of the function at 0 is 1, and the value of the third derivative of the function at 0 is 1, and so on and so forth.  Now, keep in mind that I'm not showing you all of them, because I'll be here forever then.  The fourth derivative of the function at x of e to the power x is e to the power x itself, and the fourth derivative of the function at x equal to 0 is, again, 1.  So all the values of the function at 0 and all its derivatives at that particular point are simply 1.  So what I'm going to do is I'm going to substitute it back in here.  So I'm going to say f of . . . is 0, h is like this, the value of the function at 0, plus the derivative of the function at 0 times h, plus the second derivative of the function at 0 divided by factorial 2 times h squared, plus third derivative of the function at 0 divided by factorial 3 times h cubed, plus all the other terms which are not being written there.  Now . . . so from here I get f of h is equal to the value of the function at 0, which I just calculated to be 1, the value of the derivative of the function I just calculated it to be 1 at x equal to 0, the value of the second derivative also I calculated 1 at x equal to 0, so I get that, and then I have the third derivative is also 1 divided by factorial 3 times h cubed, and so on and so forth.  So from here, what I get is that the function, if I want to calculate the value of the function at h, it will be nothing but 1 plus h, plus h squared by factorial 2, plus h cubed by factorial 3, and so on and so forth.  So what you are able to gather from here is that if I change my h to x, because h is just a dummy variable in this series, I can say 1 plus x, plus x squared by factorial 2, plus x cubed by factorial 3, and so on and so forth. But what is f of x?  f of x is nothing but e to the power x, because that's what we chose our f of x to be.  So that's the derivation of the Maclaurin series by using our . . . our Taylor series to calculate what e to the power x looks like.  And that's the end of this segment.