Hi everyone and welcome. Let's start developing some rules for differentiating certain functions. These rules will help us to find derivatives a little more quickly than having to go through the entire limit definition of a derivative. Let's start with an easy example. How about derivative of a constant function? If I have a constant function, let's say f of x equals c, where c is some constant, so it's equal to 1 or 2, or 3, or whatever Pi. That's a horizontal line. Nice horizontal line. y equals some constant. The slope of a horizontal line is 0. Any slope has zero horizontal line and the derivative, which is the slope of the tangent line to this line, the line itself is 0. This is a geometric argument behind why the derivative of a constant will be 0. We 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. Let's write it out in full, f of x plus h minus f of x all over h. Remember our function is constant. As I write the limit as h goes to 0, if I plug in h plus x into the constant function, I just get back the 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 two side limits as h approaches 0, but is not 0 as it's positive, you get 0 over some tiny number and non-zero number, whether it's positive or negative, you're going to get back 0 as this limit. In that case, you have the limit argument showing that the derivative of a constant is 0. This is going to be like our first rule, that once we work out these developments here, so I'll use Lambda notation to show it. Once we have shown it once, we're 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 that, just to show you that a simple rule can have funny interpretations. This lets you do things like if, there's an easy one, take the constant function 5, y equals 5. What's its derivative? What is the change of a constant function? 0, it's constant, there are no changes. What if constants can be scary? What if I had the constant function e to the Pi, to the root, two times a million, whatever? 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's 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're going to one trig identity, this is the one they know, sine squared plus cosine square root of any number is always 1. This is the derivative of the constant one in disguise. This is 0 as well. Constants can come in many forms. Just know when you're looking at 1. As we move on, let's look at some examples that we've seen also in trying to develop pattern for just general polynomials that are now non-constant. 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? How about if I have f of x equals a line? How about like mx? The derivative of a line is its slope. That's going to be m. If I have a function, I have the x squared before. If I have x squared, so you have to go back and 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. If I have x cubed as my original function, its derivative was 3x squared, we worked that out in the past as well. I'll leave this one as an exercise for you to verify. But if we had x to the fourth, I don't think we've done this. The algebra is nasty, but you can certainly do it. You get 4x cubed. Now here's a question for you. Do you recognize a pattern? If I have for you the general function and n degree polynomial x to the n, do you recognize the pattern? Stare at this for a minute if you don't see it, pause the video and look these over. But there is a pattern in this forming 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. If I have a function x to the n, the n falls in front and it becomes x to the one less. 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. Let's give it a name. The power rule will let us find. We'll say d/dx of x^n is going to be nx^n minus 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^4, x^5. Let's do some examples, if I have the function f of x equals the square root of x and we've done this one before, so you can look it up, remember where it is. But radicals are exponents , they're fractional exponents. The square root of x is the same as x^1/2. If I said to you find the derivative, yes, you can go through using the limits but remember we now have this power rule, it says exponents fall down in front, and then it's one less, so 1 minus a 1/2. What's that? That's f prime of x would then be equal to 1/2x to the, oh, sorry, I mess this up. 1/2 minus 1, subtract 1, so negative a half there it is, sorry about that. So 1/2 x to the minus 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 algebra to make the answers look the same. Negative exponent becomes 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 a square root, then you get the answer that we had before. So the power rule, which I did here, unlike what three lines, so you don't forget to subtract one, gives you the same answer as you would expect, there's one derivative here as the doing it the long way using limits. So we're trying to find shortcuts because as the functions get more complicated using the 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. 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 in two exponents. How about three, the cube root of x squared? I don't have a formula for this. Well, you have to convert it to an exponent. So in general, in calculus, whenever you see any sort of radical and you sort of root convert it to an exponent, so this becomes x squared and then to the 1/3, and then when you have power to a power, you multiply. So this is the same as two times 1/3 or x^2/3. They usually will hand it to you and radicals, just because it's kind of people they are, so get used to when you see a radical converting it to an exponent, and now if I asked you for the derivative, we use our power rule, so the exponent comes down and then it becomes 2/3 minus 1. So go off on the side if you need to still met that part up, but go and check it. So 2/3 minus 1 becomes x to the negative 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. Let's do some more derivative rules. We'd like these rules, 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 limits, they are certain limits, limits of a difference quotient, and therefore as limits, the limit law applies, limit laws apply. Remember limit laws say like limits behave super nice with lots of functions. So in particular, here's the first one some more once I call these one constants pop out , constants pop out. This is 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 is no, and this is the limit law that if you have a limit of a constant times 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, like let's say 7x squared. Now we never did this function 7x squared, but I want the derivative of 7x squared. Oh 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, no, you can bring the seven out front. Seven comes along for the ride and it just becomes the derivative of x squared and you say, oh, squared, we have that one, I know that one, so that's 7 times 2x, and then my final answer is just 14 x. You can also use the power rule, course see as well, so the conflicting with each other. If you do the power rule, you get 14x. The constant multiple rule, constants pop-out rule just says if you have a constant in front, you can bring up front as needed. That's good because some constants can get scary and ugly and deal with them. We care more about what's happening on the x. The other one is called, give the name. It's called the sum or difference rule. This one also follows from ANCE, falls from limit loss. It says the following: If you have the derivative of a function that is added or subtracted to each other, whereby subtraction is just like addition of the negative, then it is the derivative of the first one, and then plus or minus whatever you was to begin with, the derivative of the second one. You can think of this as the real specialists derivative as a linear but they distribute over addition and subtraction. Just to give you an example. If I had for you like the function f{x}, let's say 2x plus 7, well now if I want the derivative, I can take the derivative of the 2x and the derivative of the 7, add them up. You look on each term separately, so f{x} equals, so what derivative 2x access a line, so it's 2, and then I take the derivative of the 2x and then I add to the derivative of 7, derivative of a constant is 0, so this derivative would just be 0. Whenever you have a sum or difference, you look at each piece and you differentiate them, each way. Let's do example. How about we do one where it's maybe a little bit of word problem. Find the points on the curve, and I give you a cubic here, y equals x^3 minus 2x plus 1, where the tangent line is horizontal. Now I notice 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 like doing its thing cubic low to high, something like that because its positive leading coefficient crosses when x is 0, crosses 1, but who knows what else it's doing. It just has some general shapes, so the question like, where is the tangent line horizontal? if I had to guess, it probably looks something like this. There's a couple of places where the graph is gonna have some horizontal tangent lines, maybe there, maybe there, probably like two of them, either work it out and see. But wherever you have like these bumps in the graph. A question about the tangent line now notice nowhere in here does it say the word derivative, but the horizontal tangent line, that's like saying the slope is 0, the slope of the tangent line is 0, and here's where they tried to get tricky on as like slope of the tangent line, I know what that is, that's the derivative. This question is really saying, what points where is the derivative? Let's use y prime equal to 0. But now you have to understand what the derivative means in a geometric content. The derivative of the tangent line, or is it horizontally means the slope is 0. 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. The derivative of x^3 by the power rule, the exponent comes down and you get 3x^2. The derivative of minus 2x,is just minus 2. The derivative of the constant number 1 is 0. Final answer here, where the derivative is 3x^2. Now a lot of people, after you get good at taking derivatives, they so happy that I found the derivative, and then they box this up, put it onto the tree, and then you done. But of course, that's not what the question asked, they said it's like a multi-step problem, so Step 1 is to find the derivative. Step 2 that says where the horizontal? Remember this is the slope of the tangent line, I need to find where is 3x^2 minus 2 equal to 0. Now I have a server x, so I get x^2 equals two-thirds, and then if I, now 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 it's square root of 2/3. There are two values here, we have too little lumps on this graph of square root of 2/3. Go graph this thing and you can see how this actually works, and there are two answers. This is the final answer, not the actual derivative of themselves, that itself that they looking for. Keep this in mind that there going to be multi-step problems and this is a little preview of what's to come. Our goal is to find the derivative, so here it is right here. So say like this is going to be like Part 1, what do you do next? In this case, we just found y it's equal to 0. But like finding the derivative is always the first step. What do you do next? What do you do with the derivative that's going to be Step 2? Get used to this multi-step problem where Step 1 is finding the derivative. We do more examples and see you next time.
