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

## Discussion

## Video Transcript

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

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