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The binomial theorem expands open parenthesis a plus b close parenthesis to the nth

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power.

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so the binomial theorem

4
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Let me write it this way.

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Did they ever write this down this way?

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I don't know what this is.

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What's this sigma mean?

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The sigma means summation. We start with r equals zero and continue through n.

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The first term is a to the n times b to the zero power.

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The next coefficient is n choose one, the next number in Pascal's triangle.

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Okay, I'll explain this for you in a second.

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The corresponding powers are a to the n minus one times b to the first power.

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And you keep going until we get to n choose n.

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The final term is n choose n times a to the zero power times b to the nth power.

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Okay. And these are the numbers in Pascal's triangle, or whatever row of Pascal's

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triangle.

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The combination is n choose r, sometimes written n C r.

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It equals n factorial divided by r factorial times open parenthesis n minus r close

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parenthesis factorial.

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Have we seen this?

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What's the exclamation mark mean?

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The exclamation mark means factorial.

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Okay and

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n factorial is n times n minus one times n minus two, continuing down to one.

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times n minus 2 times dot dot dot

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times n minus

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and

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Which is 0 no, which is sorry all way down to 1

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For example, 10 choose...

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Let's go 10 choose 1.

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Ten choose one is ten factorial over one factorial times nine factorial.

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The common factorial factors cancel.

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by 1 times 9 times 8 times 7 times 6 times 5.

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The expression simplifies to ten.

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We will use the binomial theorem to prove the power rule.

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The power rule says that the derivative of x to the n is n times x to the n minus

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one.

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to any whole number power is n times x to the n minus 1. We subtract 1.

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f prime of x is the limit as h approaches zero of f of x plus h minus f of x, all

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over h.

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f of x plus h minus f of x

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all over h

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...

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The binomial expansion begins x to the n plus n x to the n minus one times h.

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The next term has coefficient n choose two and powers x to the n minus two times h

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squared.

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or the each row x to the n minus 2 h squared plus dot dot dot and then we get

48
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The final term is h to the nth power.

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Well I didn't even finish right. Yeah, so this is the last term.

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And then... so that is all this.

51
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And then we got minus x to the n

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all over h.

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The x to the n terms cancel, and every remaining term has a factor of h.

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Factor out h from the numerator.

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h to the first plus dot dot dot and then get h to the n minus 1 and then these

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simplify to 1

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Now take the limit as h approaches zero. Every remaining term except the first

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contains h.

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Those terms approach zero, leaving n times x to the n minus one.

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This proves the power rule for whole-number powers.

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whole number powers. To prove it for all values of n, we need some other stuff.

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we're not going to worry about that right now. So that's the proof of this.

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Alright, so let's do some examples.

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f of x is x.

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Oh, sorry, so I kind of mentioned this.

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This is true no matter what number this is.

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It does not have to be a whole number.

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I only proved it for whole numbers.

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We can do it for any number we want.

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But let's start with some basic examples.

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X to the 20th.

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The derivative of x to the twentieth is twenty x to the nineteenth.

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We threw that in our head.

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20x to the 19th dollar.

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Is this a rational function?

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Yeah.

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Yeah, is it a polynomial?

78
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Definitely not.

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Is it a power function?

80
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Yeah.

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This rational function is also the power function x to the negative fourth power.

82
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It's definitely a power function X the negative 4 X to any power is called a power

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function

84
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Since we know that we can find the derivative

85
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By the power rule, its derivative is negative four x to the negative fifth power.

86
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Negative 4 minus 1

87
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Equivalently, write the derivative as negative four over x to the fifth.

88
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typically write rational functions. x to the fifth.

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What type of function is the square root of x?

90
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It's a square root function.

91
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We have to call it a radical function.

92
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Is this radical function also a power function?

93
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It is.

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Rewrite square root x as x to the one-half power.

95
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We can use the power rule.

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Its derivative is one half times x to the negative one-half power.

97
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Which we typically write it like this.

98
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Equivalently, the derivative is one over two square root x.

99
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For five x cubed, first recognize the constant coefficient.

100
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it to its power function representation then use a power rule why well

101
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The expression is not just x to the n; it is five times x cubed.

102
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to the n it's 5x to the n so I can't use that rule the rules only for x to a

103
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I could use it on x cubed. I can't use it on 5x cubed

104
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Is the derivative of a constant times a function equal to the constant times the

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derivative of the function?

106
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of the function?

107
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Is that true?

108
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We prove the rule from the definition of the derivative.

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How do I start it?

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Begin with the limit as h approaches zero.

111
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We always have to start with the limit.

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As h approaches zero.

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Use f of x plus h minus f of x, all over h.

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Substitute f of x equals c times g of x.

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The numerator becomes c g of x plus h minus c g of x.

116
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And then what can I do with that c?

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This c and this c?

118
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Factor out the constant c.

119
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Okay, then a constant times a function inside a limit what can I do with that?

120
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Pull the constant outside the limit by the constant-multiple law for limits.

121
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The remaining limit is the definition of g prime of x.

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over h.

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And what is this right here?

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That's the definition of the derivative of what function?

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G prime of x.

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OK.

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Therefore, the derivative of a constant times a function is the constant times the

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derivative of the function.

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For five x cubed, the derivative is five times three x squared, or fifteen x

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squared.

131
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Real quick, one other rule.

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The derivative of f plus or minus g is f prime plus or minus g prime.

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This is the sum and difference rule.

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Together with the power rule, it lets us differentiate polynomials term by term.

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Anything separated by addition we take the derivative separately.

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Okay, so we just use the power rule a bunch of times.

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This rule is called the constant-multiple rule.