WEBVTT
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Okay, so we have a function X squared plus
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two x plus one. And we want to find
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the derivative of the function, meaning we want to
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find the derivative function f prime of banks. Then
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we want to determine the X values that we lose
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by taking the derivative. So, in other words
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, X values that are in the domain of F
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that air no longer in the domain of F crime
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. So in other words, points where the derivative
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is not defined points where the function is not differential
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all the different ways. We have to think about
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that concept. So let's just start. What's the
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definition of the derivative function? Well, I take
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the limit is h goes to zero of what?
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So I'm just going to write this out X plus
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h squared plus two times x plus a h plus
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one. So I'm gonna kind of put this in
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brackets and then minus x squared plus two x plus
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more all over h. And then, of course
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, just noticed that this is f of X plus
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h and then we're subtracting half of X. All
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right now we're just taking a limit. And now
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you're really seeing why we did all that work.
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Taking limits to get to this point. We weren't
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just taking limits for fun. We were really practicing
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taking derivatives because this derivative function is going to tell
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us so much about the original function. This is
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an extremely useful computation to do. It's not for
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nothing. So what do we need to do if
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we just try to plug in zero? We're going
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to get an indeterminant form. But the numerator here
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is just kind of begging us to expand out and
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see if anything cancels. And what happens is that
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a lot of things were going to cancel. So
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we're taking Limit is H goes to zero of what
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we have X squared plus two x h plus h
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Square e Got that just from expanding explicit squared plus
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two X plus to H plus one minus X squared
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. So here I am having to distribute the minus
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because I'm subtracting that whole quantity minus two X minus
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one all over pitch. Let's see how many things
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canceled. Well, X squared, minus X squared
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two X minus two X and one minus one.
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So we're left with the limit. His H goes
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to zero of two x h plus h squared plus
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two h all over h. And now look,
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we're taking the limit is h goes to zero so
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we can actually cancel one of thes h is from
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the numerator and the h from the Dominator. Because
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each term in the numerator has at least one factor
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of H in the denominator has a factor of h
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. So we cancel and we're almost there. Limit
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is h goes to zero of to X plus h
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plus to and what happens now If you let h
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go to zero, we're left with two x plus
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to that is the derivative function. It's a function
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that if I input numbers in the domain of X
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of ffx, it outputs the slope of the function
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at that point and we'll see a picture in just
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a second. Let's make the observation that ffx is
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defined for all real numbers. Similarly, F private
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X is the fact for all the numbers they're both
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polynomial. So what that means is that there are
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no points where F is defined. An F prime
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is not defined. So in other words, F
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FX is the differential function. It's differential over its
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entire dummy. So now let's look it a graph
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of these two functions. So here we go in
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blue, we have the function f of X.
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It's just a parabola. And then in red,
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we have the derivative function, and at first glance
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you may think, Oh well, this is just
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two functions plotted seemingly. Don't have any relationship,
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but look how amazing the derivative function ists look carefully
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right here at negative three, the derivative function has
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a value of negative four. What does that mean
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? That means the slope of the function at that
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point is negative for the slope of the tangent life
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and you see, is you move on. The
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function is leveling off, so the slope is getting
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less steep. But that's exactly what's happening to the
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derivative. It's getting closer and closer to zero while
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still remaining negative, and look at negative one to
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the left of negative one we have F prime is
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less than zero, the derivative, but the function
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of is decreasing, and then we have the cut
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off point right here, a negative one, and
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then to the right of negative one. We have
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that the derivative function is positive or in other words
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, F is increasing. And we have this amazing
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relationship between the function and its derivative. The derivative
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function is just telling you how the function is changing
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at negative one notice the derivative function equals zero,
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but that tells you that the function has a horizontal
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tangent. There's a horizontal tangent for my function right
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here. That means in this case, the F
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is changing from being decreasing thio increasing. So if
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you think about the position of a car, it
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means that I'm no longer backing up. I'm moving
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forward. It's a turning point. So you see
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the physical significance of that horizontal tangent. And so
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hopefully just looking at this craft, you're starting to
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see how amazing derivative functions are. Okay, so
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let's jump in and try to find the derivative function
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here, and we just start with what we know
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. The definition. The derivative cheap prime of tea
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is equal to. By definition, the limit is
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H purchase zero of square root of T plus H
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plus one minus square root of T plus one all
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over H, yeah. Okay, So we have
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practiced with limits. We try to plug in zero
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. We're going to get an indeterminant form. So
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what do we dio? Well, what we want
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to do. We have some square reads That's a
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red flag multiplied by the conjugate. So let's multiply
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by square it of t plus H plus one plus
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square root of T plus one. Now we have
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to do that on the top and bottom. We
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can't just do it for free. And why do
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we do that? Because the numerator now becomes a
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difference of squares. And so, in the numerator
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we have this squared minus the square. So T
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plus H plus one minus t plus one all over
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h times square it of T plus H plus one
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minus square or a sorry plus plus square root of
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T plus one. Okay. And what happens?
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We have t minus T one minus one. So
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we're left with a factor of H in the numerator
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that's going to cancel the factor of age in the
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denominator. So we're left with something very manageable limit
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. His H goes to zero of one over square
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it t plus H plus one plus square root of
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T plus one and what happens is H coast to
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zero. Well, this goes to square to t
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plus one. So you have to squared Keith plus
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ones in the denominator. So the derivative function just
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becomes too one over to times square root of T
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plus one. That's the derivative function. Okay,
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so what about the domain of F? Sorry,
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M g. I think it's confusing. The domain
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of the original function was the square root of people's
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one. So as long as I don't have a
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negative under the square root, I'm okay. So
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it seems like he needs to be greater than or
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equal to negative one. So negative one to infinity
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. But what about the domain of G prime?
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We still have this factor of square root of two
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plus one. So we know that, uh,
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the domain is included in this domain of gene negative
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one to infinity. But what about when t is
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negative? One when t is negative. One square
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root of T equals one is zero. So we're
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actually going to be there fighting by zero if we
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try to plug and they didn't want so the domain
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of G prime is actually minus one to infinity,
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not including myself. So what does that tell us
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? G of tea is differential everywhere in its domain
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. Accepted two equals negative one. So G f
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t is not differential at team equals negative one.
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Let's look at a graph these two functions and see
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what's going on. All right, so here is
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a graph of GFT in Blue Square to t plus
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one. And then you read We have the derivative
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one over to Times Square to keep this one,
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and the key thing to notice here is Okay,
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well, the directive is always positive. That means
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the function is always increasing another thing to observe.
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And you can see that initially, the slope of
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the function is very tall, so the derivative is
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very high, and then it levels off, so
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the derivative is is decreasing. But notice that the
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derivative actually has a vertical ascent dope, and that's
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because the function has a cusp like point. So
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this point here it was like a cusp point because
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it's kind of dive bombing in and the slope of
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the tangent line is getting closer and closer to infinity
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. But the derivative function is undefined. At T
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equals negative one, but the the actual function itself
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ISS is just equal to zero. So it's a
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really cool phenomenon here That happens where the function is
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to find it a point. But the derivative actually
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is not, because the slope at that point is
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too steep. All right, we know the drill
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. By now, we want to find the derivative
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function. So let's take the limit. His H
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goes to zero. In this case, we have
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our plus H to the two thirds. Okay?
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And then plus one and then minus. So this
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is a chav are plus h and then this is
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going to be just are to the two thirds plus
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what, All over What? H so again notice
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that if we try to plug in H equals zero
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, we're gonna have a big problem, and that's
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we're gonna get an indeterminant form. So first of
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all, notice that this does simplify a little bit
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. So this is the limit. His H goes
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to zero of so all of this is equal to
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H prime of our course. So our plus age
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to the two thirds minus art of the two thirds
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all over age. So all I did is just
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This is plus one minus one. Is that canceled
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? But what do we do here? This is
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really tricky. Okay, So what I would like
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to do is I would like to actually do something
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similar to multiplying by a contract. Okay, but
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this is not square. It's this is actually cubits
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. So this is a relatively hard example, I
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would say, but okay, that's fine. It
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happens. So what want to do here is I
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want to multiply by the following effect R plus H
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to the four thirds plus r plus h to the
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two thirds times are to the two thirds and then
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plus are to the four thirds and I'm gonna multiply
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by the same factor, the numerator and the denominator
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. And if you're wondering what I'm doing, kind
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of like how when you multiply by conjugate, I
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am forcing a difference of squares. Well, here
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I want to force a difference of cubes, so
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I'm really just using the difference of cubes formula.
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So again, you see how useful some of those
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algebraic tricks can be in calculus. And so what's
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the point? Well, if I multiply all this
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out in the numerator. I'm going to get a
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difference of cute. I'm gonna get this cute minus
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this cube. And if you don't believe me,
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just distribute it out. Or you could take my
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word for it either way. So this killed is
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just our plus h squared. So our plus edge
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squared and then this cube is also just r squared
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, and that's over H times. Well, I'm
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gonna do something a little bit lazy, but it's
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not a bad thing to do. I'm gonna call
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this start right here. So instead of writing all
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this all over again, I'm just going to call
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it star and start. Does it? Because what's
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significant here is what happens in the numerator. So
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in the numerator, we have our plus a squared
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months R squared. So if I simplify that out
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, I expand this binomial, I get r squared
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. That's too orange. It's h squared minus R
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squared all over edge times start. Okay, Now
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notice the are squares will cancel. I'm left with
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two r h plus h squared. So actually a
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factor of angel cancer. And so what am I
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left with? I'm left with still limit his h
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goes to zero. All I have on top is
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to R plus H over star. But the numerator
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is h goes to zero. Just goes to to
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our and then what about stock? What happens to
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star is h goes to zero. What we're gonna
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get art of the four thirds. Plus this is
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just part of the two thirds times harder than two
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thirds which is part of the four thirds, plus
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harder than four things. So we just have three
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times are to the four thirds which, of course
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, we're excluding are from the domain. Here are
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can't be zero but what this is is just two
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thirds times one over are to the one third and
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that's the answer. That's the derivative. A rich
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during the dysfunction. It's the domain of age is
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actually all your numbers you can put like any number
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I want into age and get a real number output
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but the domain of H prime. I lose a
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point kind of like in the last example I lose
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zero. So the domain of H prime is negative
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Infinity to zero, not including zero and then from
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zero up to infinity. So basically all real numbers
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except zero. So what that tells us is that
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age is not differential at R equals zero. What
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we'll see is that again we have another cuss.
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Okay, so here in blue, we have the
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original function h of our and then in red.
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We have the derivative function h prime a guard.
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And what we can see is that the slope as
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we go from the left is going to be going
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down to minus infinity. Notice The function is decreasing
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. The derivative is negative and getting steeper and steeper
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and steeper. Meghan and then to the right,
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the function is increasing. So the derivatives positive.
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But as we approach zero from the right, we're
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getting really, really steep from the right. So
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the derivative is going off to infinity, so we
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actually have a vertical ascent. Oh, Mexico.
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Zero for our derivative function because we have a customer
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point on our function. So take a look at
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that. See if that makes sense. We'll move
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on