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So in this problem were given this function F of
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X is X to the To the 3/2. And
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were asked to do several things. Were asked to
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find the domain of this, the derivative of using
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the function and the domain of the derivative. So
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, I want to start with the domain here,
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first of all. So the domain is 02 infinity
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. As I cannot do a negative number To the
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1/2 power. Okay. So there's my domain for
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my function. Now, definition of derivative definition derivative
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says F prime of X is equal to the limit
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as H goes to zero of F of X plus
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H minus F of X over H. All right
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. So for us then that means derogative fx is
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the limit as H goes to zero of X plus
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H two, the three house minus X to the
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three house all over H. And we can see
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real quick is that if we multiply this by X
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plus H two the three halves plus X to the
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three house over X plus H three halves plus X
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to the three halves. Right? Because what are
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we doing? We're doing a let's see, we're
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doing a minus B times A plus B equals a
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squared minus b squared. All right. So that
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means this is the limit. His page goes to
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zero of x plus age to the three halves squared
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. Well, the one half power squared cancels out
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. And so I'm left with X plus H cubed
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minus X cubed, aren't I? Because extra three
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halves squared would be Next to the 3rd. Okay
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. Age times X plus H. 23 halves plus
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X. To the three halves. All right now
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multiply. This out Limit is H goes to zero
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. Uh X cubed plus three X square at H
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plus three X. H squared plus H cubed minus
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X cubed over. H times X plus H.
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Two. The three halves Plus X. 2 3/2
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. All right. And what do we notice?
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First of all we noticed that X cubed minus X
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cubed. So that's gone next. We noticed that
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I got an H. Here in the denominator and
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H. And that term and H there and I
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can take one of those ages cancel the H.
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Is out. Okay And so I'm left with what
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I'm left with the limit as H goes to zero
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of three x squared Plus three XH plus H squared
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over X plus H. Two the three halves plus
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X to the three halves. Okay now perform the
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limit Well when h goes to zero that term goes
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to zero that term goes to zero and that H
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goes to zero doesn't it? Some left with three
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X squared over execute was X cubed. Extra three
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halves plus extra three halves. That's two X.
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to the three. Well I didn't want to write
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very well let's try that again. X. to
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the three house. So that's three over to well
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X squared over extra three halves. That's X.
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To the two minus three halves in the export up
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there. So that leaves me X to the one
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half, doesn't it? Okay. And the domain
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then is still zero to infinity. Just like we
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said above. Right, We can't do a negative
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number to the one half, unless we can't do
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the square root of a negative number. So here
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is my derivative and its domain.