AP Calculus AB — Logarithmic Differentiation and Derivatives of Logarithmic Functions
Corrected lesson transcript


## 0:00 — Where logarithmic derivatives fit

[0:05] Oh, we know how to find the derivative of all polynomials. And why do we know how to
[0:11] find the derivative? Or what rules do we use today? Power rule and then some different
[0:20] It takes care of polynomials.
[0:25] And then rational functions.
[0:31] Why are we able to do
[0:32] rational functions now? Because it's a quotient rule.
[0:38] Radical functions.
[0:42] Like y equals the square root of
[0:44] x squared plus three or something.
[0:47] both by the power rule and then because of the chain rule: polynomials, rationals,
[0:58] radicals, exponentials, okay we proved it for the function e to the x and then we
[1:10] did something the other day because of the chain rule for any exponential
[1:14] of function. And of course, what's the derivative of this?
[1:19] b to the x. What is the derivative of this?
[1:23] B to the x times ln of the base.
[1:28] All trig functions.
[1:31] We prove for sine and cosine, and then we use
[1:34] quotient rule for tangent. C tan, cos c tan,
[1:38] c tan. What functions are we still missing?
[1:43] that we don't know how to find the derivative of. We kind of did piece-wise, we just have to break it up into different pieces.
[1:53] We have not done any logarithms. And then the only other type we're going to learn as well is inverse trig.

## 2:06 — Start with the natural logarithm

[2:06] So today we're talking about the derivative of logarithms.
[2:14] So let's talk about this function, our favorite logarithm.
[2:21] What is the base of this logarithm?
[2:26] No, no.
[2:27] e.
[2:30] Yes.
[2:32] If it's just log, what's the base? 10.
[2:39] ln, the natural log, is log base e of x.
[2:42] of x.
[2:46] If we start with the definition,
[3:09] Can we manipulate this at all?
[3:22] What can we do with this?
[3:28] Uh oh.
[3:33] What is the difference of two logarithms?
[3:41] It is one logarithm, and inside we divide.
[3:45] ln of x plus h over x.
[3:58] Is that going to help at all?
[4:02] We play in zero for h. We get ln of x over x, which is ln of one.
[4:09] What is ln of one?
[4:14] 0 over 0. That didn't help much.
[4:20] We could do this.
[4:35] Now, what can I do with the coefficient of a logarithm?
[4:47] We can, because it is one over h.
[4:49] It's not a constant.
[4:51] But it is a coefficient of a logarithm.
[4:57] This can become the power of the inside.
[5:25] Did this help at all?
[5:39] Any ideas now?
[5:40] Now what if we plug in zero?
[5:46] Wait, is this an x?
[5:47] Is this an x? Sorry, is that what you were talking about? Thanks.
[5:55] As h goes to 0, this goes to x over x which is 1.
[6:04] And what does that go to?
[6:07] 1 over 0 goes to either infinity or negative infinity. What is 1 to the infinity or 1 to the
[6:20] negative infinity? This is an indeterminate form. Okay, long story short, this is not
[6:32] gonna work but it was a good review of rules of logarithms. Difference of logs
[6:39] is one log with quotient coefficients to become exponents but anyways long search
[6:47] yeah it's not gonna work so we gotta try something else let's go back to the
[7:04] So, before you learned about logarithms, what did logarithms, when did we come up with logarithms?

## 7:11 — Logarithms as inverse functions

[7:11] Logarithms are inverses.
[7:15] We knew any exponential looks like this, right?
[7:19] Some would depending on the base.
[7:27] This was a one to one function.
[7:29] What's a one to one function?
[7:34] It's a function. How do we know that there's a function?
[7:40] It's a vertical line test. Then it's a one to one
[7:44] function if it passed the vertical line test and it passes the horizontal.
[7:50] There's only going to be for every input there's only one output and never
[7:57] repeats the same y-value. Okay in every inverse function sorry I just gave away
[8:07] the answer. Every one to one function is guaranteed to have an inverse.
[8:17] If we reflect about the line it's this. This is the inverse of B to the X. And we just
[8:24] call this function a log with the base, wherever the base B is. Does this ring a bell? I don't
[8:35] No. What is an inverse functional?
[8:44] It is simply if we have some ordered pair in F, what is in F inverse?
[8:54] B, A. It's literally just all the ordered pairs.
[8:57] way. Okay. And that's why it was a reflection about the line y equals x. Okay
[9:07] and then inverses have a very very important property. What's the most
[9:13] important property of inverses? You compose a function and its inverse and
[9:24] And you end up with x whatever you started with there and vice versa
[9:37] Okay
[9:42] This is very important if
[9:45] We have to inverse something
[9:48] Okay, so our function e to the x and
[9:53] ln of x are inverse functions.
[10:02] That tells us e composed with ln of x is equal to x,
[10:15] ln of e to the x is equal to x.
[10:23] We agree with that
[10:26] Okay

## 10:34 — Derive the derivative of ln x

[10:34] So let's go back to this
[10:43] I'm gonna plug both sides of this equation into our function e to the x
[10:49] So the left-hand side becomes e
[10:53] to the y and the right hand side is
[11:01] e to the ln of x
[11:08] And e to the ln of x is x because they're inverse functions
[11:17] Okay, sometimes we call this the exponential form and call this the logarithmic form.
[11:25] Right, it's literally if y equals e to the x, remember this is how we found inverses,
[11:34] you just switch the ordered pairs, x equals e to the y, which is the same thing here.
[11:40] Okay, so this is exactly the same thing as this.
[11:51] Just written two different ways.
[11:55] Okay, now we're going to take the derivative of this using what kind of differentiation?
[12:11] implicit because now we have functions on both sides of the equation. We're using implicit
[12:17] differentiation. And we are looking right we're looking for the derivative of this which is dy
[12:26] dx. So we're going to take the derivative with respect to x here. So implicit with
[12:41] respect x. And the derivative of e to the y is e to the y times the derivative of our
[12:58] inside function y, what is the derivative of y?
[13:04] dy dx.
[13:07] Okay, that's the derivative of the left side.
[13:12] Derivative of x is either dx dx or just the number one.
[13:24] And then I solve for dy dx.
[13:27] I get 1 over e to the y and what is oops
[13:40] What is e to the y equal to? x. dy over dx is 1 over x.
[14:05] derivative of the natural log turns out to be this rational function.
[14:13] Okay, this is very important we need to memorize it.

## 14:35 — Differentiate logarithms with any base

[14:35] Okay, so now we know the derivative of the natural log. What about for any logarithm?
[14:42] Let's say y equals log base b of x. We're gonna do the same exact process, except
[14:58] without ease. What do we do first?
[15:13] Yeah we need to change it from logarithmic form to exponential form. So
[15:20] So I'm going to go B to both sides.
[15:38] B to the log base B of X is X,
[15:43] because they're inverse functions,
[15:45] composed through inverse.
[15:47] So this is the same function.
[15:50] As this so now we're going to differentiate using implicit differentiation
[16:03] Derivative of B to the Y is B to the Y times
[16:13] Ln of the base I think we talked about that at the very top
[16:24] derivative of any exponential function is the same function times ln of the base. Then
[16:35] we multiply by the chain rule by the derivative of the inside. The derivative of y is dy over dx.
[16:45] So that's the derivative of the left side.
[16:47] The derivative of the right side is 1 or dx over dx.
[16:53] And we solve for this.
[17:10] And then, B to the Y is simply a...
[17:30] And there it is.
[17:33] The derivative of any logarithm with any base is simply 1 over x times
[17:50] the natural log of the base. Which really this is the coefficient of the x but we
[17:59] usually write it like that.
[18:04] of any logarithm.
[18:16] So what is the derivative of log of x?
[18:28] What's the base? There's no base here. 10. It's gonna be 1 over x times the
[18:40] natural log of 10. Easy.
[18:55] Okay, let's do some more

## 19:18 — Apply the chain rule inside logarithms

[19:18] Okay, what role do we have these?
[19:24] We're going to use lots of the big ideas chain rule.
[19:30] We got a function sine of x plus x cubed inside another function log base 4.
[19:35] So this is my u.
[19:41] What is the derivative of log base 4 of u?
[19:49] 1 over u times ln of 4, but what is u equal to?
[20:05] sine of x plus x cubed times
[20:13] the natural logarithm of 4.
[20:16] Now we've got to multiply by the derivative of this.
[20:24] By the chain rule, the derivative is cosine of x plus 3x squared.
[20:39] We write it as one fraction.
[21:04] So
[21:06] Again, so if there's anything inside the logarithm, that's our U. It's 1 over U times
[21:13] ln of the base times the derivative of the inside U. By the chain rule. Let's try some more.
[21:44] So we got a function inside a function.
[21:48] What is the derivative of ln of u?
[21:52] One over u.
[21:56] Times the derivative of u.
[21:58] We have x to the 7th is 7x to the 6th.
[22:09] And what does that simplify to? 7 over x.
[22:19] Sorry, 7 over x.
[22:33] We could have done this one other way. How could I have changed this function? An exponent inside a
[22:42] logarithm becomes a coefficient. Now we don't need a chain rule, the constant multiple rule.
[22:54] It's 7 times the derivative of ln of x which is 1 over x and you're done.
[23:13] Most of the time in this class we're really just going to deal with the natural log to
[23:18] log base 4 or some other base. But not always.
[23:24] And you can always change bases too before you
[23:27] want it. So for like this, we could have
[23:30] used the change of base formula.
[23:34] What would have been the change of base formula? If we want to use the natural log,
[23:39] it's ln of the inside
[23:44] divided by
[23:47] ln of the base.
[23:51] And then this is just the coefficient.
[23:53] one over ln 4 times ln of sine x plus x cubed.
[23:59] Anyways, and then if you differentiate this,
[24:02] you end up with this times...
[24:05] Oops, that's a 4, right?
[24:09] Times 1 over ln of 4, and you get that same thing.
[24:16] Okay, let's try another one.
[24:30] Okay, you definitely have to use the chain rule.
[24:36] Derivative of ln u is 1 over u times the derivative of 5x plus 6.
[24:49] Five.

## 25:15 — Compare ln x and ln absolute x

[25:15] What kind of function is this?
[25:20] It's a piecewise.
[25:24] Uh oh. Anything involving absolute value is a piecewise function.
[25:37] It's going to be ln of x, if what?
[25:42] if X is greater than zero and it's going to be if X is less than zero, Ln of
[25:53] negative X, oops, get it right this inside is the definition of the absolute value.
[26:03] What is the domain here?
[26:11] All real numbers
[26:12] except for
[26:14] zero. Unlike
[26:21] this function
[26:22] just ln of x, what is the domain here?
[26:29] Uh oh.
[26:30] Only positive numbers.
[26:34] Okay, right, we talked about the graph of this already. Ln of x looks like that.
[26:50] What does the graph of this look like?
[26:55] To the right of zero, it is ln of x.
[27:00] What does this look like?
[27:05] This is like that graph like reflected on the
[27:09] y axis like in the classes. That's correct.
[27:13] And how'd you know that?
[27:15] Okay, that's a good guess.
[27:19] A negative in front of an x inside a function reflects about the y axis.
[27:26] For example,
[27:27] If you plug in negative 1 for x, you get ln of negative negative 1, which is ln of 1.
[27:38] a1, ln of 1.
[27:39] So it just basically reflects all these values.
[27:42] Anyways, so to find the derivative of a piecewise, what do we have to do?
[27:52] Take two separate derivatives.
[27:53] So let's start with this one. If x is greater than 0, then f of x which equals
[28:03] ln of x equals ln of x. And the derivative would be, derivative ln of x is 1 over x.
[28:25] Right now we have f prime of x is 1 over x if x is greater than 0.
[28:37] And then if x is less than 0, f of x equals ln the absolute value of x.
[28:49] Since x is less than zero, our function is this.
[29:07] Now when we take the derivative, we've got to use the chain rule.
[29:10] So, the derivative of ln of u is 1 over u, 1 over negative x times the derivative of
[29:23] the inside derivative of negative x, negative 1, just 1 over x.
[29:40] Since it's the same, we just write it as one thing.
[29:47] F' of x is simply 1 over x.
[29:58] And we already know that domain, all numbers except for zero, because you can't plug in
[30:04] zero there.
[30:11] So, the derivative of ln x is 1 over x.
[30:20] The derivative of ln of the absolute value of x is also equal to 1 over x.
[30:33] So what's the only, actually let me write it differently.
[30:46] Same story thing.
[30:59] So what is the difference between these two then?
[31:07] The domain is the difference.
[31:14] What is the domain of this?
[31:19] From 0 to infinity.
[31:21] The only positive numbers.
[31:25] That tells us the domain of this. There are no slopes of tangent lines to the left of zero.
[31:34] The domain here is also zero to infinity.
[31:41] This is the graph of ln x and this is the graph of its derivative, 1 over x.
[32:01] Here the domain is all numbers except for zero: negative infinity to zero, union zero to infinity,
[32:15] which is also the domain of this.
[32:30] Okay.
[32:35] Okay.
[32:36] Ln of the absolute value of x.
[32:41] Looks like this.
[32:45] and this is this which exactly the same on the right hand side over here the only difference is
[32:58] We have a different domain.
[33:00] Okay. So when we're finding derivatives, even though the derivative is 1 over X here,
[33:14] the domain is restricted.
[33:16] It's not the domain of this.
[33:18] The domain is only positive numbers because the domain
[33:20] of the function is only positive numbers.
[33:23] So that is very important. If the domain is not all real numbers for the function, the domain is not all real numbers for the derivative.
[33:32] It's maximum of whatever the domain of the function is.
[33:38] The distinction between these two is going to be very important in several months when we're talking about antiderivatives.
[33:52] What are antiderivatives? Antiderivatives are you're given a derivative and we want to find
[34:00] a function whose derivative is that. Okay, and this is gonna be very important later.
[34:09] But the main point of this is the domain of the function is also the most possible domain
[34:19] of this right when I say most possible like the square root of X the derivative
[34:28] is 1 over 2 square root x. The domain here is zero to infinity. What is the domain
[34:43] here? Zero to infinity, not including zero. Okay, so sometimes the numbers in the
[34:54] domain go down, right? Now we're missing one number. But anyways, my point was the
[35:07] domain of the function is at most the domain of the derivative. I don't know if
[35:12] I'm saying that quite right but it can be more restricted like it is here.
[35:26] Okay one more thing.

## 35:39 — Differentiate x to the 3x

[35:39] I told you your function.
[35:53] Is this a power function?
[36:00] Anyway, if that's the case, we're going to use the power rule.
[36:06] What is the definition of a power function?
[36:12] X to any power. And the power has to be any number, any real number. So is this a power
[36:22] function? Here is a variable, it's not a real number. So it's not a power function. So we
[36:35] We can't use power rule.
[36:42] Is it an exponential function?
[36:50] So for an exponential function, what does the base have to be?
[36:56] Greater than zero and not equal to one.
[37:03] Is this an exponential function?
[37:05] No. No, why?
[37:09] Because the base is not a number.
[37:14] So this is neither a
[37:16] power function or an exponential function. So we can't use
[37:21] the power rule or we can't use the rule for derivative of exponential function.
[37:26] So what in the world are we going to do?
[37:33] How do you
[37:35] Okay, so we can't use any of the rules that we know
[37:40] because it's neither of those type of functions.
[37:43] So we're gonna first manipulate the function
[37:45] or the equation and then take derivatives.
[37:50] And as we said, we're gonna start
[37:52] by taking the natural log of both sides or any logarithm.
[37:59] We always choose the natural log
[38:00] because that's the simplest derivative.
[38:03] So we take the natural log of both sides.
[38:17] And then what can I do with this?
[38:22] It becomes a coefficient.
[38:39] Now here we just got some function. We don't have to take the derivative of ln now. Here we got a product. We're going to use the product rule.
[38:49] And we know the derivative of ln of x. So since there's functions on both sides, what are we going to use?
[38:56] implicit differentiation.
[39:06] The derivative of ln y is
[39:10] 1 over y times the derivative of y, which is
[39:18] dy dx.
[39:22] Okay, that's just really the left side.
[39:27] Now, the derivative of the right side, we need the product rule.
[39:37] So, derivative of the 3x is 3.
[39:45] The
[39:45] derivative of 3x is 3, and the derivative of ln x is 1 over x.
[40:03] These happen to simplify, right? Plus 3.
[40:07] And we're trying to find dy dx, the derivative.
[40:12] So we solve for this.
[40:23] Times y.
[40:27] Except now we don't want x's and y's.
[40:33] What was y equal to? x to the 3x.
[40:40] So.
[40:53] And there it is.
[41:08] Okay, so when are we going to do this?
[41:17] First take the natural log.
[41:25] Whenever you have a variable both in the base and in the exponent.
[41:32] Because it's not a power function, it's not an exponential function.
[41:35] So this is called logarithmic
[41:40] I spell logarithmic.
[41:49] I'm blanking.
[41:53] Logarithm.
[41:57] There we go.
[42:00] Differentiation.
[42:12] Use whenever
[42:17] there is a variable
[42:27] And both the base and the exponent.
[42:41] And all you do is start by taking logs of both sides
[42:57] and then use implicit differentiation.
[43:21] Let's try another one.
[43:40] Let's see what this function looks like.

## 43:58 — A variable base and exponent

[43:58] It is kind of cool.
[44:02] Why are we missing a bunch of gaps in it?
[44:10] So, there's definitely some asymptote right here, but there is no, the function is not
[44:16] defined right here.
[44:18] It is not defined from pi to 2 pi.
[44:27] Why is that?
[44:34] What's that?
[44:38] It has to do with the
[44:42] We cannot have a negative base.
[44:51] Oh.
[44:53] We can't have negative numbers in a base.
[44:57] Like, negative numbers in a base.
[44:58] 4 to the power of x is not an exponential function. The base has to be positive.
[45:03] So long story short,
[45:07] whenever sine is a negative number, the function doesn't exist.
[45:12] That's why there's a bunch of gaps.
[45:15] So, let's find dy by dx.
[45:24] The derivative. And we're going to have to use logarithmic differentiation. Why is that?
[45:33] There's a variable in the base and there's a variable in the exponents. Okay. Can't use the power rule.
[45:44] can't use the exponential rule. So first step is take the log of both sides and
[46:00] what happens it will be do this exponent can become the coefficient
[46:12] Okay, so it's the same function as written differently.
[46:20] And now we differentiate using implicit.
[46:29] The derivative of ln of y is 1 over y times the derivative of the inside, which is dy
[46:36] dx.
[46:41] Then we've got a product rule.
[46:49] The derivative of cosine is negative sine, times ln of sine x, plus cosine x times
[47:07] the derivative of ln of sine x, which is one over sine x times cosine x.
[47:50] And then again, we're solving for dy dx.
[47:53] We multiply both sides by y.
[48:18] And then we replace Y with where it was, which was what?
[48:21] sine x raised to cosine x. There it is.
[48:36] functions. We've never even thought about before. And when do we use logarithmic differentiation?
[48:51] Variable in the base and a variable in the exponent. You have to use it or it's the only
[48:57] option. Okay you could use logarithmic differentiation for other functions. Okay for example, and

## 49:07 — When logarithmic differentiation is optional

[49:07] wouldn't do this but let's say we have this sine of x to the fourth power we
[49:19] would normally just use the chain rule.
[49:27] You get 4 sine cubed x times cosine x.
[49:34] It's easier. You could use logarithmic differentiation by first manipulating the equation.
[50:04] You have to use implicit differentiation, so 1 over y times dy dx is simply 4 times
[50:14] derivative of ln assigned.
[50:17] For ln of u would be 1 over u times the derivative of sine and cosine.
[50:31] over sine of x times y. What was y equal to?
[50:40] sine of x to the fourth.
[50:55] And then we can simplify this.
[51:00] sine of x
[51:01] divided by, or sine of x to the fourth, this becomes
[51:04] sine of x cubed.
[51:08] We have four,
[51:10] cosine of x times sine of x cubed, which is this right here. It's a long story short. You could use it for other problems we rarely ever do. Once or while it makes things easier.
[51:31] It definitely did not make it easier here, right?
[51:36] But you could use logarithmic
[51:39] differentiation. You have to use it
[51:41] whenever you've got a variable both in the base and in the exponent.
[51:47] Okay, any questions on this?