So this would be equal to 9. If you were to divide it by , which is actually the focus of this problem, so if we divide To divide it by , we have to divide by 10 again.
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So we move it over twice. So one, two times. And so now the decimal place is out in front of that first leading 9, which also should make sense. So if you divide it by , we should be a little bit less than 1. And so if you move the decimal place two places over to the left, because we're really dividing by 10 twice, if you want to think of it that way, we will get the decimal in front of the 0.
We should put a zero out here. Just sometimes it clarifies things. So then we get this right over here.
Now, one way to think about it, although I do want you to always imagine that when you move the decimal place over to the left, you really are dividing by 10 when you move it to the left. When you move it to the right, you're multiplying by Sometimes people say, hey, look, you could just count the number of zeroes. And if you're dividing, so over here, you were dividing by That's all right to do that, especially it's kind of a fast way to do it.
If this had 20 zeroes, you would immediately say, OK, let's move the decimal 20 places to the left. But I really want you to think about why that's working, why that makes sense, why it's giving you a number that seems to be the right kind of size number, why it makes sense that if you take something that's almost and divide by , you'll get something that's almost 1. And that part, frankly, is just a really good reality check to make sure you're going in the right direction with the decimal.
The 0 & 1st power
Because if you tried this 5, 10 years from now, maybe your memory of the rule or whatever you want to call it for doing it, you're like hey, wait, do I move the decimal to the left or the right? It's really good to do that reality check to say, OK, look, If I'm dividing by , I should be getting a smaller value and moving the decimal to the left gives me that smaller value. If I was multiplying by , I should get a larger value.
And moving the decimal to the right would give you that larger value. Up Next. All of that to the eighth power. And we want to find the derivative of our function f with respect to x. Now the key here is to realize that this function can be viewed as a composition of two functions. How do we do that? Well let me diagram it out. So let's say we want to start with, and I'll do it down here so we have some space. So we're gonna start with an x and what's the first thing that we would do?
If you were just trying to evaluate it given some x? Well, the first thing you would take two times that x to the third power plus five times that x squared and then minus seven. So, what if we imagined a function here that just did that first part. That just evaluated two x to the third plus five x squared minus seven for your x. So let's call that the function u. So whatever you input into that function u, you're gonna get two times that input to the third power, plus five times that input to the second power, minus seven.
And so when you do that, when you input with an x, what do you output? What do you output here? Well, you're going to output u of x, which is equal to two x to the third power, plus five x squared minus seven. And I'll do it all in one color just so I don't have to keep changing colors. So two x to the third power, plus five x squared, minus seven.
That is u of x. Now what's the next thing you're gonna do? You're not done evaluating f of x yet.
You would then take that value and then input into another function. You would then take the eight power of that value. So then, we will take that and input it into another function. Let's call that function v. And that function, whatever input you give it, and I'm using these squares just to say whatever input goes into that function. You're going to take it to the eighth power. And so in this case, what do you get? What do you end up with? Well, you end up with v of u of x. Or you could view this as v of two x to the third, plus five x squared, minus seven.
Or, you could view this as two x to the third, plus five x squared, minus seven. All of that to the eighth power and that's what f of x is.
The 0 & 1st power (video) | Exponents | Khan Academy
So as we just saw, f of x can be viewed as the composition of v and u. This is f of x.
So, if we write f of x. If we write f of x being equal to v of u of x, then we see very clearly the chain rule is very useful here. The chain rule tells us that f prime of x is going to be the derivative of v, with respect to u. So it's going to be v prime of, not x, but v prime of u of x. The derivative of v, with respect to u, times the derivative of u, with respect to x. So u prime of x. So we know a few things already. So let's, let me just write things down very clearly.
So we know that u of x is equal to two x to the third power, plus five x squared, minus seven.
What is u prime of x?