Hi everyone, and welcome. Let's start developing some rules for differentiating certain kind of functions. And these rules will help us to find derivatives a little more quickly than having to go through the entire limit definition of a derivative. So let's start with an easy kind of example, how 'bout derivative of a constant function? So if I have a constant function, let's say f of x = c, where c is some constant, so it's equal to one or two or three or whatever pie that's a horizontal line. Nice horizontal line, so y equals some constant. The slope of a horizontal line is 0, right? So any slope has zero of horizontal line and the derivative which is the slope of the tangent line to this line, the line itself is 0. So this is sort of a geometric argument behind why the derivative of a constant will be 0. If we sort of don't like arguing by proof by pictures, so if you want to work it out using the limit definition as h goes to 0. So let's write it out in full f of x + h- f of x all over h, re member our function is constant. So as I write the limit, as h goes to 0, if I plug in h + x into the constant function, I just get back to constant, then minus the original function itself is the constant over h. Now this 0 becomes the limit as h approaches 0 of 0 over h. Now this you can look at as a sort of 2 sided limits as h approaches 0, but is not 0 as it's positive you get 0 over some tiny number and non 0 number. Whether it's positive or negative you're going to get back 0 as this limit and in that case you have the limit argument showing that the derivative of a constant is 0. So this is going to be like our first rule that once we workout these developments here, so I'll use less notation to show it, once we have shorter ones were not going to show it again. We're just going to say we've done this, we've shown it so the derivative of a constant function is 0. Now, be careful the you know just to show you that a simple rule can have funny interpretations. This let's you do things like, this is an easy one, take the constant function 5, y = 5. What's its derivative? What is the change of a constant function? 0,it's constant, there are no changes. Okay, constants can be scary, what if I had a constant function e to the pi, to the root 2 times a 1,000,000 whatever, so constants are scary. They don't all have to look like 5. A number is a number is a number that is 0. But a number is a number is a number. Here is just one more for you. How about sine squared theta or sine squared x plus cosine squared of x? Now, remember from trig if you don't know one trig identity, this is the one to know. Sine squared plus cosine squared of any number is always 1, and so this is the derivative of the constant 1 in the skies, and so this is zero as well. So constants can come in many forms just know when you're looking at one. And as we move on, let's look at some examples that we've seen also when we tried to develop pattern for just general polynomials that are now non constant. So for example we have f of x equals a constant. We know the derivative is 0, f prime of x is 0. What if I have just a straight line? So how about if I have f of x equals a line? How 'bout like mx? The derivative of a line is it slope. So that's going to be m. If I have a function, I have the x squared before if I have x squared. So if you go back and sort of collect all your notes here, what we're doing is we're looking for a general pattern. The derivative of x squared we saw was 2x, and if I have x cubed as my original function, its derivative was 3x squared. We work that out in the past as well, and I'll leave this one as an exercise for you to verify. But if we had x to the 4th, I don't think we've done this. The algebra is kind of nasty, but you can certainly do it. You get 4x cubed. So now here's a question for you. Do you recognize a pattern? So if I have for you the general function and n degree polynomial x to the n, do you recognize the pattern? So stare at this for a minute. If you don't see it, pause the video and look these over. But there is a pattern this for me and you can, Prove this pattern actually will always hold true. The exponent on top falls in front, and then it becomes one less. The exponent becomes one less, so if I have a function x to the n, the n falls in front and it becomes x to the 1 less, and this is true. This is our general rule and we call this rule like, this observation is so great we call this the Power Rule. So let's give it a name, the Power Rule will let us find. So let's say d/dx of x to the n is going to be nx to the n -1. Anything with an exponent, we now have a nice way to find its derivative. This is a really nice pattern to notice because there's lots of things that have an exponent. And just as an example, let's do some examples here besides just regular x, the 4th x to the 5th. Remember, so let's do some examples, if I have the function f(x) = the square root of x. Now we've done this one before, so you can look it up and remember what it is. But radicals are exponents, they're fractional exponents. So the square root of x is the same as x to 1/2. So if I said, do you find the derivative? Yes, you could go through using the limits, but remember we now have this Power Rule. It says exponents fall down in front and then it's 1 less, so 1- 1/2. So what's that? So that's f prime (x) will then be equal to 1/2 x to the, sorry, one might miss this 1/2- 1 subtract 1 so negative a 1/2 there it is. Sorry about that. So 1/2 x to the -1/2. You say, wait, that's not what I remember getting or that's not what my notes show when I get to use some algebras to make the answers look the same. Negative exponents become positive in the denominator, so I can bring this downstairs. And I think if you go back and check your notes, if you replace the 1/2 with the square root then you get the answer that we had before,. So the Power Rule which I did here in 3 lines, so you don't forget to subtract 1, gives you the same answer as you would expect, right? There's one derivative here, as doing it the long way using limits. So we're trying to find shortcuts because as the functions get more complicated, using limit definition all the time. It's just going to get too cumbersome, too difficult to do. So whenever you see a radical just remember that it is the Power Rule in disguise you can use it as well. So let's just do one more as a more complicated kind of radical you can see. Also good practice with algebra to get used to how to change radicals into exponents. So how about 3 the cube root of x squared? So I don't have a formula for this we'll, you have to convert it to an exponent. So in general in calculus, whenever you see any sort of radical, any sort of route to convert it to an exponent. So it becomes x squared and then to the 1/3 and then when you have power to power, you multiply. So this is the same as 2 x 1/3 or x to the 2/3. They usually will hand it to you and radicals just because it's kind of people they are. So get used to it when you see a radical converting it to an exponent. And now if I ask you for the derivative, we use our Power Rule so the exponent comes down and then it becomes 2/3- 1. So go off on the side if you need to it will still maths that part up, but go and check it. So 2/3- 1 becomes x to the -1/3. You can totally leave your answer like this, there's nothing wrong with this answer. If you want to write it as a radical, which is fine as well, this becomes 2/3 then the cube root of x is downstairs. But sometimes people don't like radicals downstairs, but there's really no good reason why one is better than the other. Your calculator or any computer software will take both and you should be happy with both as well. So I personally don't have a preference, so I'll just box up this and get used to seeing a negative exponent. So there's nothing wrong with that. Okay, so let's do some more derivative rules, so we like these rules, right? They skip pages and pages of limits and algebra and all these things. So let's look at derivatives, so remember derivatives are limits. Derivatives are are limits, there are certain limits, limits of a difference quotient. And therefore as limits the limit law applies, limit laws apply. And remember limit laws say limits behave super nice with lots of functions. So in particular, here's the first one, some more once I call this one, constants pop out. Constants pop out, this technically the full name is like the constant multiple rule if you want to call it that. But here's what it says, it says if you have a derivative Of a constant times some function. The constant comes along for the ride, there's no, and this is a limit lot right that if you have a limit of constant test, something that the constant pops out, you can bring the constant out front. And nothing changes and nothing changes. So just as an example, how to see this thing if I have the function so let's say seven X squared. Now we never did this function seven X squared, but I want the derivative of seven squared. Say no, we don't have that in our catalog of derivatives. What am I going to do? Do I have to work this one out with all the limits? No, no, you can bring the seven out front. The seven comes along for the ride and it just becomes the derivative of X squared. You say X squared, we have that one. I know that one. So that's 7 times 2X and then my final answer is just 14X. You can also use the power rule, of course, and see that as well so they're not conflicting with each other. If you do the power rule, you get 14X. The constant multiple rule, the constants pop out rule just says if you have a constant in front, you can bring it up front as needed. That's good 'cause so you know constants and get scary and ugly. And we don't deal with them. We care more about what's happening on the X and that sort of thing. The other one is called, I forgot the name. It's called the sum or difference rule. And this one also follows from E-R-E-N-C-E, [INAUDIBLE]. It says the following. If you have the derivative of a function that is added or subtracted to each other. Remember subtractions, just like addition of the negative. Then it is the derivative of the first one and then plus or minus whatever your thing was to begin with the derivative of the second one, plus or minus the derivative of the second one. So you can think of this as the real expression is driven as a linear, but they distribute over addition and subtraction. So just to give you an example, something. So if I had for you the function F of X, let's say 2X plus seven. Well now if I want the derivative, I could take the derivative of the 2X and the derivative 7. So you look at each term separately, so F of X equals. So what's the derivative of 2X, that's the line so it's just two. And then I take the derivative of 2X and then I add it to the derivative of seven derivative of constant 0. So this derivative will just be 0. So whenever you have a sum or difference, you look at each piece and you differentiate them each piece, each way. So let's do an example. How about we do one where it's maybe a little bit of word problem. So find the points on the curve. I'll give you a cubic here. Y equals X cubed minus 2X plus one. Where the tangent line is horizontal. Where the tangent line is horizontal. Okay, so now notice like it's a little bit of word problem. This is a cubic, I don't quite know what it looks like, but I know it's doing its thing. Cubic [INAUDIBLE] high, whatever. Something like that, 'cause it's positive leading coefficient crosses when X is 0 crosses at one, but who knows what else it's doing. So it it just has some general shape. So the question is like where is the tangent line horizontal? So maybe if I had to guess, it probably looks something like this, so there's a couple of places where the graph is going to have some horizontal tangent lines. Maybe there, maybe there, probably two of '. I don't know either. Work it out and see, but wherever you kind of like these bumps in the graph. So a question about the tangent line there. Notice nowhere in here does it say the word derivative, but the horizontal tangent line. That's saying the slope is 0, the slope of the tangent line is zero. And here's where they try to get tricky so with the tangent line, I know what that is, that's the derivative. So this question is really saying what points, where is the derivative? Let's use y prime equal to 0. Now you have to understand what the derivative means in a geometric content. The derivative of the tangent line, horizontal means the slope is 0. All right, so let's put our rules together. I'm adding a bunch of things, subtracting, adding whatever. So I'll take the derivative of each piece and add it as I go. So the derivative of X cubed by the power rule, the exponent comes down and you get three X squared. The derivative of minus two X which is minus two, and the derivative of the constant number one is 0. So final answer here where the derivative is is 3 X squared. Now a lot of people after you get good at taking derivatives, they're so happy that I found the derivative and then they box this up, put it under the tree, and then you're done. But of course, that's not what the question is. Try it, they said it's like a multi-step problem. So step one is to find the derivative. Step two that says where is it horizontal. Okay, so remember this is the slope of the tangent line. I need to find where is 3x squared minus 2 equal to 0? And so now I have to solve for x. So I get x squared equals two-thirds. And then if I, be careful here. If I take a square root, if I the user take a square root, I have to put plus or minus. So its square root of 2 over 3. So there are two values here, there's probably two little lumps on this graph of square root of two over three. So go graph this thing and you can see how this actually works. And there are two answers. So this is the final answer, not the actual derivative itself that they're looking for, okay? So keep this one in mind that there's going to be multi-step problems and this is a little preview of what's to come. We will probably, our goal is to find the derivative, yes, so here it is right here. But then, so I'll say like this is going to be like part one. What do you do next? In this case, we just found where it's equal to 0, but like finding derivative is always the first step. What do you do next? What do you do with the derivative? That's going to be step two, so get used to this sort of multi-step problem where step one is finding the derivative. Okay, we'll do more examples. So I'll see you next time.
