WEBVTT 00:00:02.000 --> 00:00:11.201 We begin with a review of the power rule. If we start with the function f of x equals x to the n, 00:00:11.203 --> 00:00:22.400 then the derivative f prime of x would be n x to the n minus 1. If we start with a function of the 00:00:22.402 --> 00:00:33.398 form f of x equals u of x, where u is an expression in x raised to the n, then the chain rule or 00:00:33.400 --> 00:00:45.302 the general power rule, gives us the derivative f prime of x to be n times u of x to the n minus 1 00:00:45.304 --> 00:00:58.600 times u prime of x. Now we can do these same derivatives using a different notation. 00:00:58.602 --> 00:01:08.200 We could write this compactly into one expression by saying the derivative with respect to x, 00:01:08.202 --> 00:01:21.201 so that's d over dx, of x to the n is n x to the n minus 1. So this expression says the same thing 00:01:21.203 --> 00:01:27.601 that these two say in one little statement. We can do the same thing with the chain rule. 00:01:27.603 --> 00:01:37.401 We can write d/dx, which means the derivative with respect to x, of u of x, but instead of writing u of x 00:01:37.403 --> 00:01:44.800 at this point I'm just going to convert and simply use short notation of u. Now this is like knowing a 00:01:44.802 --> 00:01:55.901 function is called f of x but instead of calling it f of x you just call it f. So here I have u of x and 00:01:55.903 --> 00:02:05.600 I'm just going to call it u, so d/dx of u. So that's the derivative of u to the n can be written as 00:02:05.602 --> 00:02:14.700 n times u to the n minus 1, and then here, I want to show the derivative of u with respect to x. 00:02:14.702 --> 00:02:25.101 Instead of writing u prime of x, I'm going to write the derivative of u with respect to x. And so this notation 00:02:25.103 --> 00:02:31.898 means the same thing as this notation. This one turns out to be more convenient to use 00:02:31.900 --> 00:02:36.800 in the things to come. Alright, so this is a review of the power rule. 00:02:36.802 --> 00:02:44.802 We now turn our attention to topic 15 which is how to take derivatives of natural exponential and 00:02:44.804 --> 00:02:53.400 natural log functions. Now in applied calculus most of the derivatives that deal with exponential and 00:02:53.402 --> 00:02:59.401 log functions will be with natural exponential and log functions, so those are the ones we're going to 00:02:59.403 --> 00:03:10.001 discuss here. Alright first of all, natural exponential functions. The derivative of e to the x is e to the x. 00:03:10.003 --> 00:03:15.500 Look at the notation. This is the derivative of e to the x with respect to x and it's e to the x. 00:03:15.502 --> 00:03:23.101 Now that one doesn't take a very long time to remember. If you have e to the anything else besides x, 00:03:23.103 --> 00:03:30.900 if you have e to the x plus 1, e to the 2x, any other expression where this is anything besides just x, 00:03:30.902 --> 00:03:36.800 then you have basically what amounts to a chain rule, but instead of for the power rule, 00:03:36.802 --> 00:03:42.801 it's a chain rule for the natural exponential function. So what do we have? We have the 00:03:42.803 --> 00:03:52.801 derivative with respect to x of e to the u is e to the u times du/dx, and remember u is an expression in x. 00:03:52.803 --> 00:04:08.000 Alright let's do some examples and practice this. Example 1. If we have f of x equals e to the x, 00:04:08.002 --> 00:04:14.100 now we're switching back to this f of x notation. We can switch back and forth. It just depends on the 00:04:14.102 --> 00:04:26.000 convenience of what we're doing. If f of x is e to the x, then f prime of x is also e to the x. 00:04:26.002 --> 00:04:42.100 Second example. If f of x equals e to the 2x, we now have this rule because u is 2x. So the derivative 00:04:42.102 --> 00:04:51.901 f prime of x will be e to the u, just copied over, and then the derivative of u with respect to x. 00:04:51.903 --> 00:05:03.101 So we would rewrite e to the 2x, then the derivative of 2x is 2. Now as you practice this and your skill 00:05:03.103 --> 00:05:08.600 gets better and better what you may eventually do is get used to writing the 2 in the front. 00:05:08.602 --> 00:05:13.602 People just do that I think because it's easier than having to deal with parentheses or a dot, and so 00:05:13.604 --> 00:05:22.301 you end up with the derivative being 2 e to the 2x, but either one is fine, and we're practicing here 00:05:22.303 --> 00:05:28.401 so we're not going to simplify. So we'll just stop at this point. Alright let's look at another one. 00:05:28.403 --> 00:05:40.207 What if f of x is 3 e to the negative x? Ok, we have to pull from our format rules that we learned when 00:05:40.209 --> 00:05:51.301 we were learning how to use the power rule. The derivative is going to be 3 times the derivative of 00:05:51.303 --> 00:05:58.200 e to the negative x which would be e to the negative x, so that's e to the u, and then times du/dx, and 00:05:58.202 --> 00:06:06.102 the derivative of negative x is negative 1, and again this is unsimplified, but this is the actual derivative. 00:06:06.104 --> 00:06:19.501 Alright, let's try another one. Suppose we have f of x is x squared and then e to the x. 00:06:19.503 --> 00:06:28.701 Well what's the difference between this right here and what I have here? 00:06:28.703 --> 00:06:34.301 Well the difference is that this is a constant, and this is another function. So this derivative 00:06:34.303 --> 00:06:44.405 is going to require the product rule. f prime of x equals...Remember the product rule? 00:06:44.407 --> 00:06:56.007 First times the derivative of the second plus the second times the derivative of the first. 00:06:56.007 --> 00:07:03.800 Now here, because this is just e to the x, we didn't actually touch this rule. We stuck with this. 00:07:03.802 --> 00:07:16.800 Alright, let's move to some examples with a higher degree of difficulty. Number 5. 00:07:16.802 --> 00:07:34.200 f of x equals e to the 4x minus 1, over e to the x. Ok, we have a couple of choices here. We could 00:07:34.202 --> 00:07:42.004 separate this expression, simplify it, and then take the derivative, or we could write e to the x as 00:07:42.006 --> 00:07:48.400 e to the negative x in the numerator and use the product rule, or we could just go for the quotient rule. 00:07:48.402 --> 00:07:56.001 So since we haven't done a quotient rule, let's go for that. Let's say f prime of x is, now remember the 00:07:56.003 --> 00:08:08.700 quotient rule. Bottom term times the derivative of the top term is going to be e to the 4x times 4, 00:08:08.702 --> 00:08:19.304 and then the derivative of negative 1 is zero. Ok, let's stop and recap. Bottom times the derivative of the top 00:08:19.306 --> 00:08:42.905 minus the top times the derivative of the bottom all over the denominator squared. And again, 00:08:42.907 --> 00:08:48.502 this is a skill practice, and so we're not going to simplify which is probably a good thing at this point. 00:08:48.504 --> 00:08:53.102 If we were going to need this simplified, I would have recommended separating this first 00:08:53.104 --> 00:09:01.104 and then taking the derivatives of the two parts as a sum. Ok, another example. Suppose 00:09:01.106 --> 00:09:17.206 f of x is e to the negative 3x minus 4x, raised to the second power. Now do you see something here? 00:09:17.208 --> 00:09:23.805 Well here what you see is we're going to have to use the general power rule to take the derivative 00:09:23.807 --> 00:09:33.200 of this function. Considering this part, the inner function, squared, and then within that, we will use 00:09:33.202 --> 00:09:40.302 this rule when we get to the part that says the derivative of the inner function. So here we go. 00:09:40.304 --> 00:09:55.701 f prime of x equals, chain rule or general power rule. It's 2 times the inner function. Then minus 1, 00:09:55.703 --> 00:10:03.101 so that's 1 which I'm not going to put and then times the derivative of u, the inner part here. 00:10:03.103 --> 00:10:18.901 That would be e to the negative 3x times the derivative of negative 3x minus 4. Alright, I think 00:10:18.903 --> 00:10:28.600 it's time to go over this. Chain rule: 2, copy this over, subtract 1, and then take the derivative 00:10:28.602 --> 00:10:35.802 of the inner part. So we have e to the negative 3x times negative 3. That's the derivative of u, 00:10:35.804 --> 00:10:43.001 and then minus 4. And again we can stop at this point because all we're doing is practicing. 00:10:43.003 --> 00:10:52.905 Now let's take a look at natural log derivatives. The derivative with respect to x of l-n x is 1 over x. 00:10:52.907 --> 00:10:59.205 This one's a little different. It takes a few minutes to get used to. The derivative with respect to x 00:10:59.207 --> 00:11:08.502 of l-n u is 1 over u, so that parallels this, and then you have the usual du/dx, the derivative of u, 00:11:08.504 --> 00:11:19.704 and u of course is an expression in x. Again, let's look at some examples. Example 1. 00:11:19.706 --> 00:11:37.302 f of x equals the natural log of x, and so f prime of x equals 1 over x, and that's the only time we use this rule. 00:11:37.304 --> 00:11:45.401 If the expression here is anything besides x, any other form besides just x, we use this rule, and we 00:11:45.403 --> 00:12:02.604 have the du/dx after it. So let's go to example 2. f of x equals l-n x plus 4. Alright, so x plus 4 is u. 00:12:02.606 --> 00:12:18.805 f prime of x equals 1 over u and then du/dx. Now the du/dx is 1, so we could have just put 00:12:18.807 --> 00:12:25.203 1 over x plus 4, but we want to be sure that we don't forget to at least consider this. So for now we will just 00:12:25.205 --> 00:12:39.401 leave the 1 unsimplified. Let's look at another one. If f of x is the natural log of 3 x squared minus x plus 5. 00:12:39.403 --> 00:13:02.201 This is u, so f prime of x equals 1 over u, and then times the derivative of u with respect to x which 00:13:02.203 --> 00:13:11.701 would be 6x minus 1. Now if we were going to work further with this we would compress this term. 00:13:11.703 --> 00:13:17.702 We would put this in the numerator, and we would have this expression in the denominator. 00:13:17.704 --> 00:13:32.800 Let's look at a fourth example. f of x equals the natural log of x to the fifth, and now we step this up a notch 00:13:32.802 --> 00:13:41.604 because we have some decisions to make at this point. We can use this expression x to the fifth as u, 00:13:41.606 --> 00:13:48.804 and that would work fine, and it would give us the correct answer. But we also could rewrite this 00:13:48.806 --> 00:13:57.101 function using the properties of logarithms, and we could rewrite this as 5 times the natural log of x. 00:13:57.103 --> 00:14:06.705 So this is a slight advantage. Because we simplify first, it makes the derivative less complex after we take it. 00:14:06.707 --> 00:14:17.104 Now the derivative is a constant 5 times the derivative of l-n x. Choose this rule. So the derivative is 00:14:17.106 --> 00:14:28.249 going to be 5, there's the constant, and then times 1 over x. If you compare this to the other possibility, 00:14:28.251 --> 00:14:34.648 which is to think of this is u, then what we would have here, let's write it down here. We would have 00:14:34.650 --> 00:14:50.452 f of x equals natural log of x to the fifth, so f prime of x would be 1 over u and then du/dx which is 00:14:50.454 --> 00:15:01.651 5 x to the fourth. Look at the two derivatives. They obviously are equal. This is 5 over x. This would 00:15:01.653 --> 00:15:08.649 be 5 x to the fourth over x to the fifth which is 5 over x. I would encourage you though when you see 00:15:08.651 --> 00:15:15.945 a simplification that can be made using a log property to use it first. Now we're going to look at 00:15:15.947 --> 00:15:21.354 another example that people often get confused with this example. So let's take a look at this one more 00:15:21.356 --> 00:15:30.548 time before I erase it. This is the natural log of x to the fifth. Notice that only the x is raised to the 00:15:30.550 --> 00:15:38.950 fifth power. Ok, now that's what allows you to rewrite this expression as 5 times the natural log of x. 00:15:38.952 --> 00:15:59.449 Example 5 is f of x equals the natural log of x, to the fifth. Just over here I'm going to write l-n x to the fifth. 00:15:59.451 --> 00:16:05.751 That was example 4. That was f of x equals that. You see there's a significant difference. We cannot 00:16:05.753 --> 00:16:12.351 simplify this by just bringing the 5 around in the original function. We would not have the same thing. 00:16:12.353 --> 00:16:18.952 What we have here instead is the general power rule. We have a function raised to the fifth power 00:16:18.954 --> 00:16:32.848 which is totally different. So the derivative is going to be 5 times l-n x to the fourth times the derivative 00:16:32.850 --> 00:16:44.147 of l-n x which is 1 over x. Let's look at another version of this problem, and you can see that is not 00:16:44.149 --> 00:16:51.048 anywhere close to the derivative we got here. What we got here was 5 over x, and that's certainly not 00:16:51.050 --> 00:16:58.149 the same thing. We have this function in here. Now another thing happens if you have something 00:16:58.151 --> 00:17:08.649 of this form. Let's say you have the natural log of maybe, oh, 6x minus 1, and you have this 00:17:08.651 --> 00:17:17.350 whole thing raised to the third power. Then you're going to use the general power rule again, 00:17:17.352 --> 00:17:26.447 so you have the 3, and then you have this, and I switched. You can switch. You can use parentheses 00:17:26.449 --> 00:17:32.648 or brackets, whichever one seems to suit the situation better. So this is copying over the inner function, 00:17:32.650 --> 00:17:39.550 subtract 1, and now we have to take the derivative of the inner function. That means we have to 00:17:39.552 --> 00:17:50.850 use this rule. The derivative of the inner function is 1 over u times du/dx. So you almost feel here 00:17:50.852 --> 00:17:56.547 the chain rule is happening twice, and that's kind of what happens. You do the 3, and then the 00:17:56.549 --> 00:18:03.347 derivative of the inner function is this, and then the du/dx. So just be careful when you have 00:18:03.349 --> 00:18:09.550 an entire natural log function raised to a power because you get quite a long chain here. 00:18:09.552 --> 00:18:19.252 Ok, let's see what happens when we start combining. Example 7. Let's choose f of x to be a 00:18:19.254 --> 00:18:34.150 function that has the natural log of let's say a quotient, 3x plus 2 over x-squared minus 1. 00:18:34.152 --> 00:18:43.347 You can see immediately that this would be a complex fraction if you used this rule directly. 00:18:43.349 --> 00:18:56.051 But if you remember the laws of logarithms, I can rewrite this expression as the natural log of 3x plus 2 00:18:56.053 --> 00:19:05.452 minus the natural log of x-squared minus 1. Ok, quick review here. Because you have a quotient inside 00:19:05.454 --> 00:19:12.048 this natural log, the log property says that we can rewrite this as the log of the numerator minus 00:19:12.050 --> 00:19:17.350 the log of the denominator. Now we haven't taken the derivative yet. That's just rewriting the function. 00:19:17.352 --> 00:19:35.850 Now the derivative f prime of x is 1 over u times du/dx minus 1 over u times du/dx, and of course 00:19:35.852 --> 00:19:45.551 this could be simplified if we needed to do so. Ok, let's try something else now. Suppose we have 00:19:45.553 --> 00:20:00.149 f of x equals the square root of the natural log of 2x minus 1. This is actually similar to an earlier example. 00:20:00.151 --> 00:20:12.146 We could rewrite the original function as the natural log of 2x minus 1, and then the square root of the 00:20:12.148 --> 00:20:20.550 entire function means raise the whole thing to the 1/2 power. Now you may be tempted here to bring 00:20:20.552 --> 00:20:27.448 this 1/2 around to the front, but remember you can only do that when just this expression is 00:20:27.450 --> 00:20:33.448 raised to the 1/2, and here we have the entire function, so we go to the general power rule. 00:20:33.450 --> 00:20:52.248 f prime of x equals 1/2 times the natural log of 2x minus 1, raised to the negative 1/2, times the 00:20:52.250 --> 00:21:00.652 derivative of the inner function, which is 1 over 2x minus 1, times 2. Now this is probably a good place 00:21:00.654 --> 00:21:07.550 to stop and check. We rewrote the function. We did not do anything here in terms of the derivative. 00:21:07.552 --> 00:21:15.352 This is f of x, f of x, no prime. All I did was use the notation here to get it in the general power rule form. 00:21:15.354 --> 00:21:26.480 Now I actually took the derivative. I got 1/2 inner function copied, minus 1, 1 over u, du/dx.