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So we're continuing on with the exercises from 3.9.
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This is going over Question five. But again,
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before we go over any of these problems, you
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need to be very familiar with these differentiation rules.
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You've had quite a bit of time in practice with
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them, hopefully by now, so you can quickly
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look over them. But you've got your constant rule
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so derivative of a constant is always zero rules for
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addition subtraction, multiplication division. If I've got multiple
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functions in a row as well as if I have
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any sort of variable that has exponents as well And
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then what got added into this chapter was our exponential
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in log arrhythmic functions. So looking at the derivative
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of the function which is still just e the derivative
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of a power with a variable for an exponents,
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we end up having that same power again. But
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then we had a natural log of the base looking
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at even the derivative of a natural log that's one
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over X. We can also do the derivative of
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a natural log with a function. Then we follow
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our chain rules and we work outside ins. Do
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we take the derivative of the natural log than the
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derivative of the function as well as looking at the
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derivative of a log, we end up with a
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rational value, one over whatever exes. And then
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again, another natural log of the base which could
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also be thrown in with a function as well.
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So you could have the log of a function,
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and again you'd have to work through your chain rule
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as well as applying the derivative of the log.
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So looking at this problem here, we're dealing with
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a natural log where we also have a function inside
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. So we're gonna have to deal with the chain
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rule here and which, if you remember, right
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, that's when I have multiple functions. Your F
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function would be your log. Your GI function will
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be your nine X squared minus eight, and then
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we would work outside in. And then we're also
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dealing with the derivative of your log. So going
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over that again, your basic derivative for any sort
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of natural log function is won over. Whatever the
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value of X is so applying the chain rule here
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, it would take the derivative of our natural log
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, which we won over X, But our X
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is our g of X here. So we'd have
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won over nine X squared minus eight. And then
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we'd have to take our derivative of our inside our
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G of X function, which is our nine x
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squared minus. Eat that There. We have the
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power rule. Remember, With your power rule,
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we have, um that driven her power coming down
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, the exponents coming down out front, and then
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the power going down by one. So here,
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that, too, would go front, and then
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you'd end up with your, uh, nine x
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still. But now that's gonna become to the power
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of one. And then our minus eight would just
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disappear because it's a constant. So you'd end up
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with just that two x two times nine x to
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the power of one. So then cleaning this up
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, which we can a little bit, we'd still
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have one overnight X squared, minus eight. And
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here we have two times nine. So we'd have
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18 x so you can write it like that,
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leaving it side by side. Or if you wanted
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to, you could write the 18 x on top
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and have your nine x squared minus eight on the
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bottom because they're multiplying, were just multiplying straight across
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. So either way, it's a okay answer.
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Ah, you can't simplify this anymore here because we'd
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have to try to either factor of the bottom and
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see if something cancels or that's really our only option
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for going further with this question. So this would
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be your best answer to this type of problem.