WEBVTT 00:00:01.000 --> 00:00:07.200 Now that you know that the derivative is the instantaneous rate of change of a function f at a point x 00:00:07.202 --> 00:00:12.301 and you know that one interpretation of the derivative is that it is the slope of the tangent line 00:00:12.303 --> 00:00:18.299 to a curve at a point and you know how to find the derivative using the definition and you know 00:00:18.301 --> 00:00:26.100 how long that takes, it's time to find some shortcuts. So let's take a look at topic 5, derivative rules. 00:00:26.102 --> 00:00:32.501 You may have noticed a pattern in earlier work with using the definition to find the derivatives, so 00:00:32.503 --> 00:00:44.840 let's go back and review this. You may remember that if you have the function f of x equals 5x minus 7 00:00:44.842 --> 00:01:04.001 that the derivative, f prime of x, was 5, and if you had f of x to be 2 x squared minus 4x plus 6, 00:01:04.003 --> 00:01:13.601 the derivative, f prime of x, was 4x minus 4. We found these using the definition. 00:01:13.603 --> 00:01:28.200 Let's look at one more. If f of x is 3, then f prime of x is 0. So what's the pattern? 00:01:28.202 --> 00:01:36.401 Well first of all, these three functions have something in common. They all belong to a class of functions 00:01:36.403 --> 00:01:45.201 where the function is either written, or can be written, as x to the n. So for example, this is 5 x to the one, 00:01:45.203 --> 00:01:53.401 and this can be thought of as 7 x to the zero. This is x squared. This is 4 times x, but it's still x to the one. 00:01:53.403 --> 00:01:57.900 This is x to the zero, x to the zero. So what happens in the derivative? 00:01:57.902 --> 00:02:06.800 Well here the constant appears to disappear, and the derivative is the coefficient of the x-term. 00:02:06.802 --> 00:02:14.400 Here the derivative looks like it's something to do with the 2 and the 2 here. The power goes down by 1, 00:02:14.402 --> 00:02:22.301 and then this x is gone. So you see a pattern. These are all examples of the power rule used in combination 00:02:22.303 --> 00:02:28.100 with some other rules. So let's take a look at what the power rule actually says. 00:02:28.103 --> 00:02:36.801 The power rule. If a function f of x has the form x to the n, now it either has to have this form or be able 00:02:36.803 --> 00:02:45.800 to be written in this form, where n is a real number, then the derivative f prime can be found by 00:02:45.802 --> 00:02:55.600 writing n, then x to the n minus 1. So in effect what do you actually do? It looks like you bring the power 00:02:55.602 --> 00:03:01.000 around to the front and you subtract 1 from the power that was there. So lets take a look at some 00:03:01.002 --> 00:03:17.500 simple examples of the power rule. Alright, number one: f of x equals x-squared. So according to the 00:03:17.502 --> 00:03:24.900 power rule, the derivative, not using the definition but using this rule would be to bring the 00:03:24.902 --> 00:03:34.901 2 around to the front, recopy the x, and then n minus 1 would be 2 minus 1 which is 1, 00:03:34.903 --> 00:03:40.000 which we're actually not going to write. So the derivative of x squared is 2x. 00:03:40.002 --> 00:03:50.801 This is going to become very familiar. Let's look at another one. If f of x equals x cubed, then according 00:03:50.803 --> 00:04:00.300 to the power rule the derivative f prime of x would be... now think through how you're going to do this. 00:04:00.302 --> 00:04:08.801 You're going to bring the 3 to the front and then rewrite the x and then subtract 1. So we're using the power rule. 00:04:08.803 --> 00:04:15.401 Let's look at some other basic functions that either are power or can be put in the form of a power rule. 00:04:15.403 --> 00:04:22.700 So what if you have just x? Well, what you do is you think of this as x to the 1. Ok now if you 00:04:22.702 --> 00:04:32.400 imagine a 1 here, then the derivative f prime of x would be, bring the 1 around to the front and then 00:04:32.402 --> 00:04:41.300 write the x and then 1 minus 1 would be 0. Ok if you recall any number raised to the zero power is 1, 00:04:41.302 --> 00:04:49.000 so what we have is 1 times 1, so what we're going to actually write is 1. So let's go back and look. 00:04:49.002 --> 00:04:54.300 The derivative of x squared is 2x, and we're not going to really write this 1. 00:04:54.302 --> 00:05:09.400 The derivative of x cubed is 3 x squared, and the derivative of x is 1. Alright, now let's go further 00:05:09.402 --> 00:05:20.801 and look at some other basic functions. The derivative of f of x equals 1 over x. Ok we have a little 00:05:20.803 --> 00:05:29.700 problem here. This is not in the form x to the n, so what we have to do in order to use the power rule, 00:05:29.702 --> 00:05:34.501 because that's all we have at this point unless we want to go back to the definition with the limit, 00:05:34.503 --> 00:05:42.701 is we have to rewrite this as x to the negative 1. Then we take the derivative using the power rule. 00:05:42.703 --> 00:05:51.101 So this is going to be negative 1 around to the front, then x. Now subtract 1, so this would be 00:05:51.103 --> 00:05:59.601 negative 1 minus 1 or negative 2. Now that is the correct derivative. If you want to simplify this, you can. 00:05:59.603 --> 00:06:07.400 It depends on your needs. Right now we're doing basically skill and drill, and at that point you can stop 00:06:07.402 --> 00:06:13.200 and use that for your answer. If you need to work further with it, you could always rewrite this as 00:06:13.202 --> 00:06:21.301 negative 1 over x-squared. It just depends on the need. Alright, let's take a look at another one. 00:06:21.303 --> 00:06:33.101 Alright, f of x equals the square root of x. Again it's not in power form, but we can easily put it in the form 00:06:33.103 --> 00:06:45.401 of x to the n. We'd write it as x to the 1/2, so the derivative f prime of x equals, power rule, 00:06:45.403 --> 00:06:57.498 1/2 x to the negative 1/2, and again that is sufficient unless you need for some reason to simplify it. 00:06:57.500 --> 00:07:09.701 You could always write it as one over 2 square root x. Ok, let's go one more. So if f of x equals, let's 00:07:09.703 --> 00:07:18.401 use say something with a negative power, so maybe our original function is x to the negative 1/4. 00:07:18.403 --> 00:07:36.142 It's still the form x to the n, and so the derivative f prime of x would be negative 1/4 times x to the, 00:07:36.144 --> 00:07:41.340 now you subtract 1. When you subtract 1, you're going to have to get a common denominator, so what 00:07:41.342 --> 00:07:48.945 you actually have is negative 1/4 minus 4/4. You get used to converting these pretty fast, and 00:07:48.947 --> 00:08:00.242 you get negative 5/4 for the exponent. Again, you could simplify this if you needed to, but for now that 00:08:00.244 --> 00:08:18.040 is sufficient. Alright, let's take a look at another example. Let's use f of x equals 1 over the cube root of x. 00:08:18.042 --> 00:08:31.940 Now this is not in power form, but we can write this as x to the negative 1/3. So when we take the 00:08:31.942 --> 00:08:44.240 derivative using the power rule, we would have negative 1/3 times x to the, and then we're going to 00:08:44.242 --> 00:08:53.543 subtract 1 from this exponent, so that would be 3 over 3, and so we would have negative 4 over 3. 00:08:53.545 --> 00:09:02.940 Alright, now not everything is going to be exactly in this form, so let's take a look at another rule 00:09:02.942 --> 00:09:11.440 that is a spinoff of the power rule. This is called the constant rule, and it says if f of x is k, where k is 00:09:11.440 --> 00:09:19.639 a constant, then the derivative of a constant is 0. Now if you remember, we started off at the beginning 00:09:19.641 --> 00:09:27.739 talking about a function, several, but one of them was f of x equals 3, and we knew the derivative was 0 00:09:27.741 --> 00:09:36.439 because we found the derivative using the definition in the topic before this one. So we know the 00:09:36.441 --> 00:09:46.039 derivative is 0. Let's look at how this could possibly be a spinoff of the power rule. This is 3, and it's 00:09:46.041 --> 00:09:55.339 really times x to the 0. If you use the power rule, it would be 0 times 3 is 0 times this term which 00:09:55.441 --> 00:10:01.638 ends up being 0, but you don't actually do this every time you take the derivative of a constant. 00:10:01.640 --> 00:10:11.140 Instead you simply just remember the derivative of a constant is 0. Alright, let's see. What if our function is 00:10:11.142 --> 00:10:24.938 f of x equals 0? Well 0 is a perfectly good constant, and the derivative of 0 is 0. It seems a little strange, 00:10:24.940 --> 00:10:36.930 but that's what it is. So what if we have a combination of these? Well we have another rule that's a spinoff. 00:10:36.932 --> 00:10:41.538 This one is constant times a function rule. Again it's a format rule. We're still going to have to use 00:10:41.540 --> 00:10:46.040 the power rule to actually get the derivative or some other rule that will come up later. 00:10:46.042 --> 00:10:53.839 So here's the relationship. If the function has k, a constant, times a function, then the derivative is 00:10:53.841 --> 00:10:58.640 that same constant times the derivative of the function. So it's actually very easy to handle. You 00:10:58.642 --> 00:11:05.340 pull the constant out to the front. You take the derivative of the function, and then you put it back together. 00:11:05.342 --> 00:11:18.139 So here's what it looks like in practice. You have f of x equals something like 3 x squared. Alright, 00:11:18.141 --> 00:11:26.720 so what you're going to do here is you're going to think of this as 3 times x squared, and you're going to 00:11:26.722 --> 00:11:36.540 take the derivative of x squared using the power rule. That would be 2x. So we would have the constant, 00:11:36.542 --> 00:11:51.738 which is this k right here, but it's a 3, and then the derivative of x squared is 2x, and so we get 6x. 00:11:51.740 --> 00:11:59.240 So what actually happens is mechanically you multiply the power times the constant in the front, 00:11:59.242 --> 00:12:07.639 write it down here, and then subtract 1 from the power of the variable. Let's look at another one. 00:12:07.641 --> 00:12:25.040 Ok, let's try f of x equals 1 over 2x. We're going to have to write it in another form so that we can 00:12:25.042 --> 00:12:30.740 use the power rule. So we're going to leave the 1/2. Then we're going to write this as x to the negative 1. 00:12:30.742 --> 00:12:37.639 So now we have a constant times a function, and that function is of the form x to the n, so the power rule applies. 00:12:37.641 --> 00:12:54.405 So the derivative, f prime of x, is negative 1 times the 1/2, so that's negative 1/2 and then x to the negative 2. 00:12:54.407 --> 00:13:04.006 We could simplify this if we needed to. For now it is just fine. Ok, what happens if we have a function of this type and 00:13:04.008 --> 00:13:10.704 then maybe added or subtracted to another function of a similar type, or that could be written as a similar type? 00:13:10.706 --> 00:13:17.200 Well, we have another spinoff rule. This one is called the sum and difference rule, and it says 00:13:17.202 --> 00:13:25.803 if your original function is the sum or the difference of two functions, then you can simply find f prime by 00:13:25.805 --> 00:13:30.800 taking the derivative of the first one plus or minus, whatever the original one was, 00:13:30.802 --> 00:13:37.200 and then the derivative of the second one. It's very easy, and it's exactly what you would like it to be. 00:13:37.202 --> 00:13:43.705 So let's look at example 12 which would be an example of how to use the sum and difference rule. 00:13:43.707 --> 00:14:00.704 Let's have f of x equal 6 x squared plus 4x to the negative 2. So we can look at this as 00:14:00.706 --> 00:14:07.303 one function added to another function. So we're going to take the derivative of this one, 00:14:07.305 --> 00:14:11.401 then put a plus sign and the derivative of this one. So what do we get? 00:14:11.403 --> 00:14:24.600 Well, we get f prime of x equals, 2 times 6 is 12, and then subtract 1 from the 2, so we get x, 00:14:24.602 --> 00:14:31.103 carry over the plus sign. Then derivative of this side. Now at this point you realize that you have 00:14:31.105 --> 00:14:39.502 negative 2 times 4 which is negative 8, so instead of putting the plus sign here you look ahead, 00:14:39.504 --> 00:14:44.204 and if you do, you see that what you're going to have is negative 8, and you go ahead and write that 00:14:44.206 --> 00:14:51.302 right there and then x to the negative 3 because we're subtracting 1 using the power rule. 00:14:51.304 --> 00:14:58.701 Alright, so that's what we do if we have either a sum or a difference. Let's do one more of these. 00:14:58.703 --> 00:15:16.805 Number 13: f of x equals, and let's try something like this. 5 times the cube root of x minus 10x. 00:15:16.807 --> 00:15:29.099 Alright, now remember that this is x to the 1/3. Let's rewrite this. 5 x to the 1/3 minus 10x. 00:15:29.101 --> 00:15:34.401 Now this is still f of x. I have not taken the derivative, so I don't write f prime yet. 00:15:34.403 --> 00:15:42.600 That's a common error so be careful on that. So what is the derivative? This might be a good time to stop 00:15:42.602 --> 00:15:50.202 and appreciate the rules and realize how unpleasant it would be to use the definition of the derivative here, 00:15:50.204 --> 00:15:54.802 where you'd have to have a limit as h goes to 0 of f of x plus h and so on and so forth. 00:15:54.804 --> 00:16:05.605 You get an appreciation about this time. The derivative would be 5/3 x, now we're going to subtract 1 here. 00:16:05.607 --> 00:16:16.903 The 1 that we subtract is 3/3, so we have 1/3 minus 3/3 which is negative 2/3, and then the derivative 00:16:16.905 --> 00:16:27.504 of negative 10x is negative 10. One more time, why is it that? Because this is x to the 1, and so when 00:16:27.506 --> 00:16:33.204 I bring the 1 around, multiply by the 10, I get 10 times x to the zero which is 10 times 1 which is 10. 00:16:33.206 --> 00:16:43.755 So now we know what to do if we have the sum or the difference of two functions. 00:16:43.757 --> 00:16:49.800 Now what we'd like to do is we'd like to have a rule this simple for a product of two functions. 00:16:49.802 --> 00:16:55.304 It doesn't really work that way though. Unfortunately, it does not give you the correct derivative. 00:16:55.306 --> 00:16:59.503 It doesn't represent the slope of the tangent line to a curve at a point or the instantaneous 00:16:59.505 --> 00:17:06.903 rate of change of the function. So we have another rule that actually gives you the right derivative. 00:17:06.905 --> 00:17:12.604 Here's what happens if you have the product of two functions. Now we have to make up some words here 00:17:12.606 --> 00:17:20.102 that are easy to remember, so for lack of anything better, we call this one the first and this one the second. 00:17:20.104 --> 00:17:29.103 When you multiply you can't just memorize g of x and h of x because sometimes the functions have 00:17:29.105 --> 00:17:37.600 different names. So we go first and second. Alright, so what is f prime? Well here it is. 00:17:37.6022 --> 00:17:48.104 It's the first function, copied over, times the derivative of the second, then plus the second function 00:17:48.106 --> 00:17:54.204 times the derivative of the first. Ok, so you get this little sing-song rule that you memorize. 00:17:54.206 --> 00:18:01.805 First derivative second plus second derivative first, and that's the product rule. Alright, example time. 00:18:01.807 --> 00:18:15.699 Number 14: f of x equals, let's start out with something relatively simple, 4 x squared minus x 00:18:15.701 --> 00:18:28.300 times x to the fifth minus 1. We could just multiply because neither of these is raised to a power. 00:18:28.302 --> 00:18:33.303 Well, they're both raised to the 1 power, so we could actually use FOIL and multiply this out. 00:18:33.305 --> 00:18:50.300 But let's try the product rule. Alright, f prime of x is the first, simply copy this over. Don't touch the inside, 00:18:50.302 --> 00:18:57.100 just copy it over, and then the derivative of the second. Alright, this is the second. When we take the 00:18:57.102 --> 00:19:02.703 derivative of the second, we're going to use the power rule because really that's the only rule we really have 00:19:02.705 --> 00:19:12.200 that actually gives us the derivative. So this would be 5 x to the fourth, bring this around subtract 1, 00:19:12.202 --> 00:19:24.100 and then the derivative of a constant is 0. So what we have here is first, derivative of the second, 00:19:24.102 --> 00:19:36.103 plus second and then the derivative of the first, which would be 8x minus 1. 00:19:36.105 --> 00:19:51.600 8x minus 1. Let's go back over why this is minus 1. This is actually negative 1 [times] x to the 1. 00:19:51.602 --> 00:19:59.200 When you bring the 1 around it's 1 times 1 x to the 0. x to the 0 is 1. 1 times 1 is 1. What happens is 00:19:59.202 --> 00:20:04.899 you never figure that out each time you do it. You quickly remember that the derivative of x is 1. 00:20:04.911 --> 00:20:12.403 So let's check the rule again. First derivative second plus second derivative first. 00:20:12.403 --> 00:20:14.500 First 00:20:14.502 --> 00:20:15.301 Derivative second 00:20:15.303 --> 00:20:16.803 plus second 00:20:16.805 --> 00:20:19.005 Derivative first. 00:20:19.007 --> 00:20:31.003 Ok, let's look at another one. Example 15: f of x equals 00:20:31.005 --> 00:20:51.699 6 x cubed plus 2 x squared minus 5, so this is a trinomial, times the square root of x plus 1 over x. 00:20:51.701 --> 00:20:56.900 Now this one is going to be particularly nasty when we get to the part that says derivative of the 00:20:56.902 --> 00:21:01.597 second because we're going to have to take the derivative of this. So the first thing I'm going to do is 00:21:01.599 --> 00:21:10.204 just make a mental note that this is actually x to the 1/2, and that this is x to the negative 1. 00:21:10.206 --> 00:21:28.398 That's going to help when I take the derivative. Alright, product rule. f prime of x equals first 00:21:28.400 --> 00:21:37.799 copied over, no changes, times the derivative of the second, and again, slowly. Derivative of the second, 00:21:37.801 --> 00:21:48.097 1/2 x to the negative 1/2. I brought the 1/2 around to the front. I subtracted 1 here. Then plus 00:21:48.099 --> 00:21:54.501 because this is the sum of two functions, except that when you look ahead you see that when you bring 00:21:54.503 --> 00:22:00.798 the negative 1 around this is going to be negative. So just go ahead and fix it here. It makes the work look 00:22:00.800 --> 00:22:09.399 a lot neater. So this would be negative 1 times x to the negative 2. Now at this point you often get 00:22:09.401 --> 00:22:14.803 a little distracted. The fact that you're using the product rule sometimes escapes you because you're 00:22:14.805 --> 00:22:20.102 working so hard here on the power rule and the sum and difference rule. So let's go back and 00:22:20.104 --> 00:22:29.003 look at what we have. This is first. This is derivative of the second. Then plus the second. 00:22:29.005 --> 00:22:34.703 Now when you copy this over you can copy it over originally or you can copy it over with these powers, 00:22:34.705 --> 00:22:43.803 whatever pleases you at the moment. I'm just going to go ahead with the 1/2 and the negative 1 00:22:43.805 --> 00:22:50.802 since that's what I actually used. So we have first derivative second plus second, copied over, 00:22:50.804 --> 00:23:06.602 derivative of the first, 18 x squared plus 4x. Then what's the derivative of negative 5? 00:23:06.604 --> 00:23:18.002 Zero. So we have first derivative second plus second derivative first. Now this is a skill problem. 00:23:18.004 --> 00:23:22.002 We're not going to make any attempt to simplify this or multiply this together. 00:23:22.004 --> 00:23:26.501 We're trying to learn a rule. So that's the product rule. 00:23:26.503 --> 00:23:33.299 Alright, let's move on to what happens if we have the quotient of two functions. Well, we have a quotient rule. 00:23:33.299 --> 00:23:41.199 The quotient rule looks like this. If a function is g of x over h of x, which is one function divided by another, 00:23:41.201 --> 00:23:48.500 hence the quotient, the derivative is... and before I read this to you we have to come up with some shortcut. 00:23:48.502 --> 00:23:54.202 Now we all know that this is the numerator and this is the denominator, but those words are very long, 00:23:54.204 --> 00:24:01.001 and so we take a little shortcut here. We call this one the top and this one the bottom. So the derivative, 00:24:01.003 --> 00:24:08.502 f prime, is bottom times derivative of the top minus the top times the derivative of the bottom, 00:24:08.504 --> 00:24:13.999 over the bottom squared, and you get used to it. You just memorize this little sing-song rule. 00:24:14.001 --> 00:24:17.799 Bottom derivative top minus top derivative bottom over bottom squared. 00:24:17.801 --> 00:24:24.004 Alright, time for an example. I think it's safe to say at this time that you can tell that 00:24:24.006 --> 00:24:32.004 you need to practice this if we're doing this many examples. Alright, so let's take f of x equals 00:24:32.006 --> 00:24:46.201 4 x cubed minus 6x over 3 minus the square root of x. So we recognize here that there is a quotient, 00:24:46.203 --> 00:24:52.203 and in order to take the derivative, it is not so simple as to take the derivative of the numerator 00:24:52.205 --> 00:24:57.602 divided by the derivative of the denominator. It doesn't give you the right derivative. So instead 00:24:57.604 --> 00:25:07.899 we use this nice little rule here. Here we go. It's bottom, now when I copy this over I'm going to write 00:25:07.901 --> 00:25:16.602 it as x to the 1/2, so that's bottom. Why would I do that? Because when I get to the part that says 00:25:16.604 --> 00:25:21.199 derivative of the bottom, I'm going to already have it in the right power form to use the power rule. 00:25:21.201 --> 00:25:34.701 So I have bottom, derivative of the top. And what did I do? I was so busy concentrating on this I 00:25:34.703 --> 00:25:38.699 forgot to take the derivative of the top, and this is pretty standard. You have to realize there are a lot of 00:25:38.701 --> 00:25:43.899 pieces to this problem, and you always go back and check what you're doing. So you have bottom. 00:25:43.901 --> 00:26:04.204 Let's try again, derivative of the top. Ok, bottom derivative top minus top derivative of the bottom. 00:26:04.206 --> 00:26:08.502 The derivative of the bottom. Now instead of looking at this, I'm going to just look at this one. 00:26:08.504 --> 00:26:20.800 The derivative of 3 is 0. This would be, using the power rule, it would be the negative 1/2 x to the, 00:26:20.802 --> 00:26:33.301 subtract 1, negative 1/2. Now this is where a lot of people stop. Don't stop here. Go back and check 00:26:33.303 --> 00:26:39.900 your work, and you realize that we still have this little part down here, bottom squared. You can use 00:26:39.902 --> 00:26:49.201 either form. I guess it would be consistent if I used the 1/2, put that in parentheses, and squared it. 00:26:49.203 --> 00:26:59.403 Alright, this is the quotient rule. Let's check it. Bottom derivative of the top minus 00:26:59.405 --> 00:27:10.803 the top times the derivative of the bottom. That would be 0 and then that, over the bottom squared. 00:27:10.805 --> 00:27:16.001 We will not simplify at this point. You really actually would never want to simplify this until 00:27:16.003 --> 00:27:21.900 you really had to, and so that is the derivative using the quotient rule. 00:27:21.902 --> 00:27:35.406 Alright, let's take a look at one more situation. Sometimes you have a function that's a composite function. 00:27:35.408 --> 00:27:41.402 So let's just take a look at what one would look like before we go the actual rule. You could have something 00:27:41.403 --> 00:27:50.505 like this. You could have like a 4 x squared minus x, but then you could have that function raised to a 00:27:50.507 --> 00:27:59.000 power, let's say negative 2. Now if it was raised to the positive 2 power you could just square it and then 00:27:59.002 --> 00:28:04.199 take the derivative using the sum and difference rule like we've done earlier, but when it's raised to the 00:28:04.201 --> 00:28:12.405 negative 2 power, there is nothing you can do to expand this term, so we're going to need another rule. 00:28:12.407 --> 00:28:21.806 But if you look at this it has some of the characteristics of x to the n, the power rule. So what does it have 00:28:21.808 --> 00:28:29.405 in common? Well it has something to a power, but instead of just having x to a power, it has this 00:28:29.407 --> 00:28:38.401 whole function raised to a power. Now this is a lot like what I have on. You could think of this as 00:28:38.403 --> 00:28:44.503 all one function, or you could think of it as the inner function, which is the 4 x squared minus x, 00:28:44.505 --> 00:28:49.902 and then the outer function, which is raise it to the negative 2 power. So you really have a function 00:28:49.904 --> 00:28:58.304 raised to a power. So instead of x to the n, we have to come up with a name or something to call this 00:28:58.306 --> 00:29:05.403 before we identify it in a rule. The common dummy variable that's used is the letter u. 00:29:05.405 --> 00:29:10.204 So instead of calling this f of x, that wouldn't work because we've already used this here, and we 00:29:10.206 --> 00:29:20.904 could use g of x or we could use h of x, but it's very common to use u of x for this. So the u of x is going 00:29:20.906 --> 00:29:33.402 to be the inner, and then the n is the negative 2. So what we need now is a rule to show us what to do 00:29:33.404 --> 00:29:40.800 when we have this inner part. It follows the pattern of the power rule, but as you can expect there is 00:29:40.802 --> 00:29:45.602 more to it than that. So let's look at the rule, and then we'll do some examples. 00:29:45.604 --> 00:29:50.904 It's called the chain rule because when you do it you feel like you're chaining things together. 00:29:50.906 --> 00:29:57.304 Actually it's just a generalized form of the power rule, and you could see that. It looks like the power rule, 00:29:57.306 --> 00:30:05.501 but it's got this inner function. Here's what you have to check. First of all, u of x has to be differentiable, 00:30:05.503 --> 00:30:13.400 so that's this part in here, and n has to be a real number just as it did with the power rule, and you have 00:30:13.402 --> 00:30:19.202 to have this form which is exactly what I developed right here, inner function and then the power. 00:30:19.204 --> 00:30:27.702 In that case the derivative, not surprisingly, is n, which is the power brought around to the front, then 00:30:27.704 --> 00:30:34.504 times the u of x, which means you just copy the inner function over, then you subtract 1, but then you have to 00:30:34.506 --> 00:30:41.406 remember there is another part to this. You then have to take the derivative, u prime of x, 00:30:41.408 --> 00:30:51.100 of the inner function, and you see you feel the chain. It's like this is connected to this. Alright, example time. 00:30:51.102 --> 00:31:00.602 Let's start out with something that is relatively simple to work with just so we can get the rule straight, 00:31:00.604 --> 00:31:14.810 and then we'll bump it up another notch. So let's use 6 x squared minus 3x, raised to the 4th. Now we 00:31:14.812 --> 00:31:22.502 actually could, if we were just bored and had nothing else to do today, we could raise this to the 4th power. 00:31:22.504 --> 00:31:27.601 But there are two problems with that. First of all it's boring and tedious, and second of all, 00:31:27.603 --> 00:31:32.202 then you won't learn how to use the chain rule. So let's use the chain rule. 00:31:32.204 --> 00:31:43.600 f prime of x is n times the function, the inner function, n minus 1, and then the derivative of the inner. 00:31:43.602 --> 00:31:57.303 So you get 4 times 6 x squared minus 3x, subtract 1, and then don't stop. Feel that there is a chain here. 00:31:57.305 --> 00:32:11.200 You now have to take the derivative of the inner part using the power rule. 12x minus 3, and again 00:32:11.202 --> 00:32:16.300 we are not going to simplify, and so that's all we really have to do. Alright, let's do one more. 00:32:16.302 --> 00:32:40.206 Number 18: f of x is x to the negative 2 plus 4, raised to the 3rd, and let's combine this. Let's multiply this by, 00:32:40.208 --> 00:32:51.102 oh let's say, 3x minus 8. Now what have I done here? Well, I've got a combination of several rules going here. 00:32:51.104 --> 00:32:59.518 I've got a product. I have a function times a function, but the first function is a u of x to the n and is 00:32:59.520 --> 00:33:06.419 going to require the chain rule when I get to the part that says derivative of the first. Ok, so let's remember 00:33:06.421 --> 00:33:18.620 this is the first, don't forget about the 3. This is the second. So our overriding format rule is the product rule, 00:33:18.622 --> 00:33:24.019 but the chain rule is going to be hidden in it, along with the power rule. So here we go. 00:33:24.021 --> 00:33:38.220 First times the derivative of the second, which is just 3. The derivative of a constant is 0. 00:33:38.222 --> 00:33:52.319 So we have first derivative second plus second derivative of the first. Whoa. The derivative of the first 00:33:52.321 --> 00:34:00.720 is going to require the chain rule, so it's going to be 3, that's the [exponent] brought around to the front, 00:34:00.722 --> 00:34:14.519 copy over the inner function, subtract 1, and just because I ran out of space here doesn't mean you 00:34:14.521 --> 00:34:22.219 can forget there's another part to this, so multiplied by this is the derivative of the inner function, 00:34:22.221 --> 00:34:32.020 the u of x. The derivative of this would be negative 2 brought around to the front, x to the negative 3, 00:34:32.022 --> 00:34:41.320 and then plus the derivative of 4 is 0. So this part is multiplied here. Alright, let's do another one of these. 00:34:41.322 --> 00:34:52.020 Let's make both the first and the second raised to a power other than one, so we have to use the 00:34:52.022 --> 00:35:02.920 chain rule twice. Alright, let's put in something like x to the fourth minus x, raised to the negative 2. 00:35:02.922 --> 00:35:14.019 So you see we're going to need a chain rule here. And then times 3x minus 1, to the fourth. 00:35:14.021 --> 00:35:25.320 The product rule is going to be the format, but we're going to have to use the chain rule 00:35:25.322 --> 00:35:29.919 every time we take the derivative of the first or the second. So here we go. 00:35:29.921 --> 00:35:47.722 First times the derivative of the second which is 4 times 3x minus 1, subtract 1, and then the 00:35:47.724 --> 00:35:55.819 derivative of the inner which is 3. So let's go back and check this before we finish. This is first, 00:35:55.821 --> 00:36:11.819 derivative of the second. Then plus the second copied over times the derivative of the first, which 00:36:11.821 --> 00:36:25.919 uses the chain rule. Negative 2, x to the fourth minus x, raised to the, subtract one here, negative 3. 00:36:25.921 --> 00:36:38.918 What do I do last? The derivative of the inner function, 4 x cubed minus 1. Now, don't stop at 00:36:38.920 --> 00:36:49.722 this point. Go back and check your work. We have first, then the derivative of the second using the 00:36:49.724 --> 00:36:59.318 chain rule. You end up with four parts here. Then plus the second, then times the derivative of the 00:36:59.320 --> 00:37:06.520 first, and again there are your four parts, and that is the derivative of the original function f of x. 00:37:06.522 --> 00:37:13.219 Ok, just to be sure we didn't lose sight of what was going on, this function represents the 00:37:13.221 --> 00:37:19.619 instantaneous rate of change of this function, at x. It also represents the slope of the tangent line 00:37:19.621 --> 00:37:27.620 to the graph of this function, at x. It is a whole lot easier to use this rule, or this combination of rules, 00:37:27.622 --> 00:37:33.222 to get the derivative than it is to use the definition. Now we have reached a point where 00:37:33.224 --> 00:37:38.082 we can actually begin to use the derivative for something.