AP Calculus AB — The Power Rule and Basic Derivative Rules
Corrected lesson transcript


## 0:00 — The Power Rule

[0:05] The binomial theorem expands open parenthesis a plus b close parenthesis to the nth power.
[0:13] so the binomial theorem

## 0:15 — The Binomial Theorem

[0:15] Let me write it this way.
[0:36] Did they ever write this down this way?
[0:40] I don't know what this is.
[0:43] What's this sigma mean?
[0:46] The sigma means summation. We start with r equals zero and continue through n.
[0:58] The first term is a to the n times b to the zero power.
[1:08] The next coefficient is n choose one, the next number in Pascal's triangle.
[1:19] Okay, I'll explain this for you in a second.
[1:23] The corresponding powers are a to the n minus one times b to the first power.
[1:28] And you keep going until we get to n choose n.
[1:33] The final term is n choose n times a to the zero power times b to the nth power.
[1:40] Okay. And these are the numbers in Pascal's triangle, or whatever row of Pascal's triangle.
[1:49] The combination is n choose r, sometimes written n C r.

## 1:58 — Combinations and Factorials

[1:58] It equals n factorial divided by r factorial times open parenthesis n minus r close parenthesis factorial.
[2:11] Have we seen this?
[2:12] What's the exclamation mark mean?
[2:14] The exclamation mark means factorial.
[2:18] Okay and
[2:21] n factorial is n times n minus one times n minus two, continuing down to one.
[2:24] times n minus 2 times dot dot dot
[2:29] times n minus
[2:32] and
[2:33] Which is 0 no, which is sorry all way down to 1
[2:40] For example, 10 choose...
[2:46] Let's go 10 choose 1.
[2:53] Ten choose one is ten factorial over one factorial times nine factorial.
[3:09] The common factorial factors cancel.
[3:21] by 1 times 9 times 8 times 7 times 6 times 5.
[3:30] The expression simplifies to ten.

## 3:40 — State the Power Rule

[3:40] We will use the binomial theorem to prove the power rule.
[3:45] The power rule says that the derivative of x to the n is n times x to the n minus one.
[4:01] to any whole number power is n times x to the n minus 1. We subtract 1.

## 4:13 — Prove the Power Rule

[4:13] f prime of x is the limit as h approaches zero of f of x plus h minus f of x, all over h.
[4:20] f of x plus h minus f of x
[4:26] all over h
[4:29] ...
[4:42] The binomial expansion begins x to the n plus n x to the n minus one times h.
[4:54] The next term has coefficient n choose two and powers x to the n minus two times h squared.
[5:09] or the each row x to the n minus 2 h squared plus dot dot dot and then we get
[5:23] The final term is h to the nth power.
[5:39] Well I didn't even finish right. Yeah, so this is the last term.
[5:47] And then... so that is all this.
[5:54] And then we got minus x to the n
[6:01] all over h.
[6:13] The x to the n terms cancel, and every remaining term has a factor of h.
[6:31] Factor out h from the numerator.
[6:46] h to the first plus dot dot dot and then get h to the n minus 1 and then these simplify to 1
[7:14] Now take the limit as h approaches zero. Every remaining term except the first contains h.
[7:24] Those terms approach zero, leaving n times x to the n minus one.
[7:36] This proves the power rule for whole-number powers.
[7:45] whole number powers. To prove it for all values of n, we need some other stuff.
[7:54] we're not going to worry about that right now. So that's the proof of this.

## 8:06 — Positive Integer Powers

[8:06] Alright, so let's do some examples.
[8:10] f of x is x.
[8:12] Oh, sorry, so I kind of mentioned this.
[8:18] This is true no matter what number this is.
[8:22] It does not have to be a whole number.
[8:23] I only proved it for whole numbers.
[8:26] We can do it for any number we want.
[8:29] But let's start with some basic examples.
[8:36] X to the 20th.
[8:42] The derivative of x to the twentieth is twenty x to the nineteenth.
[8:50] We threw that in our head.
[8:52] 20x to the 19th dollar.

## 8:54 — Negative Integer Powers

[8:54] Is this a rational function?
[8:58] Yeah.
[8:58] Yeah, is it a polynomial?
[9:03] Definitely not.
[9:07] Is it a power function?
[9:09] Yeah.
[9:09] This rational function is also the power function x to the negative fourth power.
[9:13] It's definitely a power function X the negative 4 X to any power is called a power function
[9:20] Since we know that we can find the derivative
[9:27] By the power rule, its derivative is negative four x to the negative fifth power.
[9:35] Negative 4 minus 1
[9:38] Equivalently, write the derivative as negative four over x to the fifth.
[9:44] typically write rational functions. x to the fifth.

## 9:49 — Fractional Powers

[9:49] What type of function is the square root of x?
[10:00] It's a square root function.
[10:02] We have to call it a radical function.
[10:06] Is this radical function also a power function?
[10:10] It is.
[10:11] Rewrite square root x as x to the one-half power.
[10:15] We can use the power rule.
[10:22] Its derivative is one half times x to the negative one-half power.
[10:40] Which we typically write it like this.
[10:45] Equivalently, the derivative is one over two square root x.

## 10:47 — Coefficients Need Another Rule

[10:47] For five x cubed, first recognize the constant coefficient.
[10:51] it to its power function representation then use a power rule why well
[11:14] The expression is not just x to the n; it is five times x cubed.
[11:25] to the n it's 5x to the n so I can't use that rule the rules only for x to a
[11:38] I could use it on x cubed. I can't use it on 5x cubed

## 11:45 — The Constant-Multiple Rule

[11:45] Is the derivative of a constant times a function equal to the constant times the derivative of the function?
[11:56] of the function?
[11:58] Is that true?

## 11:59 — Prove the Constant-Multiple Rule

[11:59] We prove the rule from the definition of the derivative.
[12:02] How do I start it?
[12:05] Begin with the limit as h approaches zero.
[12:06] We always have to start with the limit.
[12:08] As h approaches zero.
[12:11] Use f of x plus h minus f of x, all over h.
[12:30] Substitute f of x equals c times g of x.
[12:47] The numerator becomes c g of x plus h minus c g of x.
[12:57] And then what can I do with that c?
[13:01] This c and this c?
[13:02] Factor out the constant c.
[13:09] Okay, then a constant times a function inside a limit what can I do with that?
[13:17] Pull the constant outside the limit by the constant-multiple law for limits.
[13:20] The remaining limit is the definition of g prime of x.
[13:29] over h.
[13:31] And what is this right here?
[13:37] That's the definition of the derivative of what function?
[13:43] G prime of x.
[13:46] OK.
[13:48] Therefore, the derivative of a constant times a function is the constant times the derivative of the function.
[13:54] For five x cubed, the derivative is five times three x squared, or fifteen x squared.

## 14:00 — The Sum and Difference Rule

[14:00] Real quick, one other rule.
[14:04] The derivative of f plus or minus g is f prime plus or minus g prime.
[14:22] This is the sum and difference rule.
[14:26] Together with the power rule, it lets us differentiate polynomials term by term.
[14:29] Anything separated by addition we take the derivative separately.
[14:34] Okay, so we just use the power rule a bunch of times.
[14:36] This rule is called the constant-multiple rule.