AP Calculus AB — Derivatives of Sine, Cosine, and e^x Corrected lesson transcript ## 0:00 — Derivatives of Sine, Cosine, and e^x ## 0:05 — Warm-up: A Tangent Line from the Power Rule [0:05] Alright, let's start with a little warm-up before we get into actually the new material. [0:20] Find the equation of the tangent line at x = 0. [0:38] X equals zero. [0:45] To find the equation of the tangent line, we need its slope. [0:51] To find the slope we need to find the derivative. [0:54] We will use several derivative rules. [1:04] We're not going to use the definition. [1:06] What's the first rule? [1:09] Since we have a sum of a bunch of power functions, we're going to use the... [1:15] No. [1:16] Sum of... [1:18] The sum... [1:19] Did we name it? [1:20] Did I call it something? [1:21] Did I write it down, but we didn't call it something? [1:28] Use the sum and difference rule. [1:31] The derivative of a sum is the sum of the derivatives. [1:37] For each power function, apply the constant multiple rule and the power rule. The derivative of 5x cubed starts with 3 times 5, or 15. [1:49] Subtract one from the exponent to get the second power. [1:52] These are easy enough, hopefully we can just do it in our head, we don't have to write [2:06] The derivative of 7x is 7, and the derivative of the constant is 0. [2:18] The derivative of a cubic is quadratic; the degree decreases by one because of the power rule. [2:22] 1 because the power rule and okay here's the derivative at any value x so we want [2:29] Evaluate the derivative at x = 0. [2:41] The slope of the tangent line is 7. Now find its equation. [2:51] The y-value is f(0) = -2. [3:00] Use point-slope form: y minus y-one equals m times x minus x-one. [3:22] The tangent line is y = 7x - 2. [3:27] Okay, so the power rule is very nice, yes? [3:31] It's a lot simpler than, [3:33] we had to use the definition of the derivative [3:34] on that thing and take us a year to do so. [3:42] And of course the sum and difference rule [3:44] and the constant multiple rule. [3:47] Okay, so that was kind of review from Wednesday. ## 3:50 — Derive the Sine Rule [3:50] Today, let's talk about, [3:57] Now find the derivative of sine. [4:05] How are we going to do this? [4:08] Umm, there's rules. [4:11] There's rules? [4:15] I don't know of any rules. [4:17] Negative sign? [4:20] Oh my goodness. [4:24] Now we're going to do it. [4:26] How are we going to find the derivative? [4:30] Thank you. [4:32] Use the definition of the derivative. [4:35] Which is? [4:39] Start with the limit as h approaches zero. [4:42] As h approaches zero. [4:44] Use f(x+h) minus f(x), all over h. [4:53] This becomes sine of x plus h minus sine of x, all over h. [5:15] Okay, now what? [5:18] Distribute sine? [5:23] What do you mean? [5:28] Do something with this? [5:31] What can we do with that? [5:33] Sine of x plus h is not sine x plus sine h. [5:44] Oh. Let's do an example. [5:47] Pi over 2 plus pi over 2. [5:53] What's pi over 2 plus pi over 2? [5:56] Sine of pi. [5:59] Is that sine of pi over 2? [6:02] plus sine of pi over 2. Sine of pi is 0. What is sine of pi over 2? 1. Does 0 equal 2? [6:18] Definitely not. Okay, we definitely can't do that. Uh oh, we need something else. Any other [6:27] What are ideas? [6:40] Can I factor out a sign? [6:45] You get x plus h minus x. [6:48] Definitely can't do that. [6:50] There's no such thing. [6:53] Use a trigonometric identity. [6:55] This is very important. [6:56] Who said that? [6:57] I said that one actually. [6:58] You did? [6:59] Good. [7:02] Okay. [7:41] I don't know, it's not an awesome thing. [7:44] It's awesome. [7:46] Okay, uh, so, we gotta... [7:49] These are some of the trigonometric identities we will use. [7:52] There are infinitely many trigonometric identities; these are the main ones for this lesson. [7:54] These are the main ones. [7:56] Which one are we gonna use? [7:58] Use the sine sum identity. [8:01] What'd it go? [8:04] What? [8:05] Uh oh. [8:07] Let's go back here. [8:11] The argument contains a sum, so use the sine addition identity. [8:17] Sum a difference. [8:18] Sum a difference. [8:19] Use sine of alpha plus beta. [8:38] That identity is sine alpha cosine beta plus cosine alpha sine beta. [8:46] the plus and the minus means if it's plus then this is plus if it's minus then it's [8:53] negative so [8:56] Got [9:06] With alpha = x and beta = h, it becomes sine x cosine h plus cosine x sine h. [9:13] sine x cosine h plus cosine x sine h. [9:38] Okay, now what? [9:47] Any ideas? [9:51] What? [10:00] Group the sine x terms and the sine h terms. [10:09] that so first let's rearrange it [10:22] Rearrange the numerator. [10:26] Factor out sine x and cosine x. [10:31] This guy and this guy. [10:53] I like that. [10:55] Now what? [10:58] Um, cosine h minus 1 is identity of something. [11:08] That's it. [11:09] That's cosine square. [11:10] I forgot what these are. [11:11] They have to be cosine squared. [11:12] Oh, that's square. [11:13] They're not. [11:14] They're not. [11:14] We have to be cosine squared minus 1. [11:17] Cosine square. [11:20] Uh, any other ideas? [11:23] this. [11:35] Split the expression into two limits. [11:37] This plus that. [11:39] This becomes the limit of sine x times cosine h minus one, over h, plus the limit of cosine x times sine h, over h. [11:58] The first special limit equals zero. [12:07] Sine x is constant with respect to h, so it may be moved outside the limit. [12:12] You don't have to do this, but you could bring it outside the limit. [12:16] The cosine-minus-one limit equals zero. [12:23] The sine-over-h limit equals one. [12:28] The result is sine x times zero plus cosine x times one. [12:37] And there it is! [12:40] The derivative of sine x is cosine x. [12:43] Beautiful. [12:49] That is math has finds. [12:51] Please prove that derivative of sine is cosine. [12:55] So are we gonna do this every single time? [13:00] Definitely not. [13:01] At this point, we just memorize. [13:06] Derivative of sine. [13:10] Okay next one. ## 13:12 — Derive the Cosine Rule [13:12] Now find the derivative of cosine. [13:30] Let's work it out people! [13:33] How are we going to start? [13:36] Begin with the limit as h approaches zero of f(x+h) minus f(x), all over h. [13:40] f of x plus h minus f of x all over h. [13:59] Okay. I bet you could do it. [14:03] Finish the proof or finish the limit. [15:59] What's the next step? [16:06] Let's switch these guys. [16:46] Separate the expression into two limits. [16:55] This goes to zero. [17:00] The cosine-minus-one limit is zero and the sine-over-h limit is one, giving cosine x times zero minus sine x times one. [17:14] The result is negative sine x. [17:16] Okay, and I think when we went over graphing derivatives we graphed cosine, we graphed [17:22] When we graphed the derivative of cosine, it also matched negative sine x. [17:27] We proved it. Beautiful. [17:30] Okay, so we we memorize this one [17:34] at this point [17:37] It's negative sine of x [17:40] The derivative of sine is cosine, and the derivative of cosine is negative sine. [17:46] All right, now we can do any polynomial. [17:51] Any multiple of sine, any multiple of cosine. ## 17:55 — Why Tangent Must Wait [17:55] Now consider tangent. [18:40] If we substitute this in, it's going to be a really ugly mess. [18:48] The definition with tangent of x plus h would be a complicated expression. [18:52] Okay, so we're not going to do this [18:56] This right here is going to be real messy [19:04] Rewrite tangent as sine x divided by cosine x. [19:15] Now what [19:19] That's as far as I got. [19:21] I like this. [19:26] The derivative of a sum is the sum of derivatives. [19:39] Is that another rule? [19:45] It's definitely not. [19:49] Okay. [19:51] The derivative of a quotient is not the quotient of the derivatives. [19:56] It's true for sums and differences, not true for quotients. [20:00] Okay. I don't know what I just did here. [20:03] Long story short, we have no idea what the derivative is. [20:06] We will learn the quotient rule after the test. [20:14] We haven't proved that yet, right? [20:20] So that's for next week after the test. [20:29] We cannot finish tangent until we have the quotient rule. [20:34] We're going to do one more. [20:39] No more trig. [20:40] The rest are all quotients. [20:42] We don't have any more quotients. [20:43] Right, uh, cosecant is one over sine, which is another quotient. [20:55] We don't have a quotient rule yet. [20:59] We're going to do e to the x. ## 21:15 — Where the Number e Comes From [21:15] Before we start, what is our number e? [21:18] The number e is approximately 2.71828; it is irrational. [21:39] is the real definition of E? Where did it come from? Natural log is the inverse of E [21:51] to the x. So we first had to find what E to the x was for natural log. Anybody remember [21:59] when we first talked about it? I know we did it at Math 2.3 several years ago. [22:12] Where do we use this in math 2.3? [22:19] Compound interest. [22:23] Okay. [22:24] Compound interest uses A = P times the quantity 1 plus r over n, raised to the nt power. [22:25] It's a principle, A equals principle. [22:33] Principle times E. Let's do compound interest first. [22:41] The factor is 1 plus r over n, raised to nt. [22:47] Oh boy, people. [22:49] N plus E plus. [22:51] Yeah, that's fine. [22:51] R is your interest rate. [22:52] The variable n is the number of compounding periods per year. [23:06] larger we got the more it just kept getting closer to this one number and [23:15] And then we came up this formula for continuously compounded interest [23:22] E to the RT, right? [23:25] Okay, it has everything to do with this the real definition of E [23:32] The number e is the limit as n approaches infinity. [23:41] X goes to infinity [23:46] Use the quantity 1 plus 1 over n, raised to the nth power. [23:56] which is just that compound [23:58] This is the compound-interest limit with the extra rate and time parameters removed. [24:04] As you go to infinity, this gets very, [24:07] very small, but then the power gets very, [24:09] very big at the same rate. [24:12] We're at the same time, [24:13] It turns out this is exactly the definition of E. This is where the E comes from. [24:20] There is an equivalent definition. [24:27] As h approaches zero, use the quantity 1 plus h, raised to the 1 over h power. [24:40] So, 1 over x. [24:47] These two definitions are equivalent. [24:51] As x goes to infinity, 1 over x goes to 0. [24:56] And as x goes to infinity, this goes to infinity, 1 over x goes to infinity. [25:00] sorry, one over zero goes to infinity. [25:04] Okay, we're gonna use, well, we need this, [25:09] except we're gonna call it this instead. [25:13] So keep this in mind. [25:17] Doesn't matter what the variable is, same exact thing. [25:23] Okay, we'll come back to that. ## 25:26 — Derive the Exponential Rule [25:26] Now let f(x) = e to the x. [25:31] The derivative is the limit as h approaches zero. [25:37] Use f(x+h) minus f(x), all over h. [25:38] minus f of x all over h. [26:00] Okay, now what? [26:13] Factor the numerator after using exponent rules. [26:16] That's awesome. [26:16] Okay, I like that except what do we got to do with this first? [26:26] Rewrite e to the x plus h as e to the x times e to the h. [26:29] You have addition in the exponent. We can change this to e to the x times e to the H [26:42] Factor out e to the x. [27:03] I [27:03] Like it now what? [27:08] Replace e using its limit definition. [27:13] Use e as the limit of the quantity 1 plus h, raised to the 1 over h power. [27:22] zero, one plus H to the one over H. What we just talked about. [27:38] Can we have limits inside limits? [27:42] Absolutely. [27:42] be. [28:12] Okay, now what? [28:21] Use the power law for limits to bring the exponent into the inner expression. [28:26] Is a limit of uh, let me just get the law [28:36] I think this was the limit law the limit is whatever x approaches any number [28:44] f of x to the any number n is n to the limit. [28:57] This was one of the limit laws. [29:00] So basically we're going to bring this inside the limit and put it over the function. [29:37] Ok, now what? [29:48] For a power raised to a power, multiply the exponents. [29:52] What can we do? [29:53] We can't solve it. [29:54] We can't solve it. [29:55] We can't solve it. [29:58] Rule of exponents any base with an exponent all raised to an exponent. What do we do the exponents times them? [30:06] multiply [30:06] The exponent 1 over h times h simplifies to 1. [30:39] Now what? [30:45] The limit of the constant 1 is 1. [30:51] Use the sum and difference laws for limits. [30:55] so we can bring this negative one inside here. [31:12] One plus h minus one simplifies to h. [31:20] Okay. [31:23] Now these ones add to be zero. [31:28] Okay. [31:30] And let me bring this guy out here. [31:31] Factor e to the x outside the remaining limit. [31:37] And this is the limit as h approaches 0. [31:49] The ratio h over h equals 1. [31:53] 1. [31:54] 1. [31:56] This is 1. [31:58] This is 1. [32:00] We get e to the x times 1. [32:05] Therefore, the derivative of e to the x is e to the x. [32:11] All this for this? [32:14] Yes, it was awesome. [32:17] We just proved the derivative of e to the x is e to the x. [32:22] It's the easiest derivative to find. [32:27] The definition-based proof depends on the limit definition of the number e. [32:32] definition of the number E. Okay and we probably really didn't tell you that [32:43] that's the definition because you didn't know what limits were back a couple years ago. [32:47] But now we do. That is the definite. So long story short, we've memorized this [32:53] From now on, memorize that the derivative of e to the x is e to the x. [33:07] Any constant multiple of e to the x differentiates to the same constant multiple of e to the x. [33:14] Same seed. [33:16] Now that we have proved the derivative of e to the x, we can use the rule directly. [33:21] We now know derivatives of polynomials, sine, cosine, and e to the x. [33:30] Or the sum of any combination of those. ## 33:34 — Combine the New Rules [33:34] Apply the rules to a mixed example. [33:55] This is some crazy function. [33:57] who knows what it looks like so how are we gonna find the derivative here so [34:18] The derivative of a sum is the sum of the derivatives, so differentiate each term separately. [34:24] Differentiate each term separately, beginning with the power rule. [34:37] The derivative of 5x to the fourth is 20x cubed. The derivative of 2 sine x is 2 cosine x. [34:53] The derivative of negative 4 cosine x is positive 4 sine x. [35:04] The derivative of 8e to the x is 8e to the x. [35:19] The derivative is 20x cubed plus 2 cosine x plus 4 sine x plus 8e to the x. [35:27] There it is. [35:27] This gives the slope of the tangent line at any value of x. [35:35] Okay. [35:47] Okay. [35:47] from? [35:50] Okay. ## 36:10 — Product and Quotient Rule Preview [36:10] We know the derivatives of e to the x and sine x, but a product needs another rule. [36:15] Do we know the derivative of a product? [36:20] What's the product rule? [36:36] What's the product rule? [36:48] this is why in the world is it that we got to prove it maybe it's something [37:03] We need the product rule for this expression and the quotient rule for tangent. [37:12] Save those rules for next week after the test. [37:20] The rules from today are enough for the assigned Section 2.2 problems. [37:28] We will stop here.