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All right, so we're continuing to work on the
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exercises from 3.9, So this one's looking at question
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nine itself. But before you can do those,
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you need to be very comfortable with thes rules for
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differentiation. You would have been using them up until
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now. So just going through them briefly, You've
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got your directive of Constance. What happens if I
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have addition, subtraction, multiplication and division as well
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as what happens if I have functions of functions and
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how to deal with powers where I've got a variable
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base. We also added in this chapter are derivatives
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for exponential on log rhythmic functions. So what does
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the derivative of the e function look like versus a
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power with a variable exponents versus natural logs and regular
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logs? So, looking at this problem here,
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I've got a natural log I have to deal with
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as well as divisions. I'm gonna have to use
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my division rules or my quotient rules. So remember
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, with division I've got my two functions my top
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function, my bottom function, my numerator, my
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denominator. I'm gonna take the drivel of the first
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, multiply it by the regular second, subtract the
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derivative of my bottom multiplied by my regular top and
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then divide that all by my regular bottom squared.
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So I'm gonna have to deal with that division rule
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on how that plays in. And I'm also gonna
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have to deal with my natural log. So going
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through that the derivative, my natural log is just
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one over X. So those are the two that
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are gonna take place in this kind of equation,
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and we'll work through accordingly. So if I look
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at it my F function or my first function,
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my top one is my natural log. So I
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take the derivative of that which is one over X
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on Multiply that by my regular function on the bottom
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, which is just X. Then I subtract the
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derivative. My bottom, the derivative of X becomes
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technically, we move the one down and then it
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becomes one minus one, leaving it to a power
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of zero. But what happens with that X is
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I'm just left behind with the one I'd multiply that
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by my regular top, my f of X without
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the derivative. And then I have to divide this
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whole thing by my bottom functions squared. Now,
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cleaning this up a bit because we can we notice
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Here, I've got one over x times X So
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one times x x divided by ex that cancels thes
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out. So I'd be left with one there and
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then I don't have to write the one out front
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of the natural log. I can just leave.
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That is a natural log. And then I'd have
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my X squared still in the bottom. That I
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can't do anything with and not here would be how
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I would solve this. There's not much more I
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can do with that. That's vote is nice and
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clean as I can get it. And that's kind
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of how I apply my division rules with my natural
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log rules.