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Derivative Functions - Example 1

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity, and the derivative of velocity with respect to time is acceleration. The concept of a derivative has been generalized to other contexts than just functions of a single real variable. For instance, the derivative of a function of several real variables is a function of several real variables, and the derivative of a function from Rn to Rm is a function from Rn to Rm.

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

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