WEBVTT
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All right. So we want to find the second
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derivative function of this function ethics. Okay, so
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first things first. How do you find the second
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driven? The second derivative is the derivative of the
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first trip. Okay, so let's just find the
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first derivative first. So the first derivative is gonna
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be the limit. His H goes to zero of
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X plus h cubed minus three X plus H squared
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minus 24 X plus H minus seven. Okay,
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so that's FX plus age minus X cubed minus three
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X squared minus 24 x plus. So oh,
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over H. And of course, if you try
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to plug in H equals zero, you're going to
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get an indeterminant form. Unfortunately, we have to
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expand and hope a lot of things canceled in the
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new greater. So let's expand the numerator. And
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in fact, let's just expand the numerator and then
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right with the limit equals so expanding the numerator.
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So we have X cubed from this term X cubed
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. I'm just gonna use kind of the binomial theorem
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three x squared H plus three x h squared plus
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H cube. Okay, that's from this first term
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and then minus three X squared minus six X h
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minus three h squared because that's coming from this term
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. So I just multiply this finer mealtime itself multiplied
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by negative three and then minus 24 X minus 24
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age minus seven. But the minus seven we're gonna
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cancel you see that right away? Okay, so
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we just have minus execute plus three x squared plus
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24 X. Okay, so that's what the numerator
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is. It's a mess. It's very easy to
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make mistakes. So probably it's a good idea to
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write this out big and clear what I'm doing here
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and let's see what Kansas X Cubed cancels minus X
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cubed. We have minus three x squared plus three
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x squared. Then we have minus 24 X plus
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24 tracks. So all in all, what is
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the numerator become? So the derivative is the limit
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. It is a joke. Goes to zero of
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the numerator, which is three x squared H plus
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three x h squared plus H cube minus six X
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H minus three h squared minus 24 h Oliver age
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. Okay, And let's cancel one factor at age
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. So as we kind of expected everything is gonna
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have a factor of age. But now if we
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take ht zero well, three x squared. That
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doesn't happen. Age. So that's left. This
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still has a factor of age, so that goes
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to 03 x h h goes to zero here,
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H goes to zero. Here, this is just
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minus six X. So that's left this three h
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has a factor of h. So that goes to
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zero his age close to zero. And then we
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have minus 24. Okay, so that's the derivative
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. Easy enough supreme for the second derivative. We
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just take the derivative of the first trip. So
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it's still limit his h goes to zero. But
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now, instead of plugging an F of X plus
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age ffx, we're gonna plug in F prime of
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X plus H minus f prime of X. So
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this is gonna be three times X plus H squared
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minus six has expose age minus 24 minus F prime
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attacks, which is three X squared, minus six
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x minus 24. And of course, all over
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H. So you see the important thing to notice
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here. Is it the second derivative? It's just
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the derivative of the first derivative. So we're doing
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the same thing we did upon the derivative except using
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that crime instead of that. So now if we
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simplify this out again, a lot of things should
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cancel. We have three x squared plus six x
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h plus three H Square minus six X minus six
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age minus 24 minus three. Expert distribute plus six
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x plus 24 all over age. So let's do
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it. Cancels three X squared minus three X squared
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minus six X plus six x minus 24 plus 24
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We're left with six x h plus three h squared
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minus six age. We can cancel one factor of
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H and all we're left with is six x minus
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six plus three h. But his H goes to
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zero in the H goes to zero. So all
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we're left with it's six x minus six, and
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Matt is the second derivative of that function.