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Let $f(n)=n / 2$. Find a formula for $f^{(k)}(n)$. What is the value of $f_{1}^{*}(n)$ when $n$ is a positive integer?...

Question

Let $f(n)=n / 2$. Find a formula for $f^{(k)}(n)$. What is the value of $f_{1}^{*}(n)$ when $n$ is a positive integer?

Let $f(n)=n / 2$. Find a formula for $f^{(k)}(n)$. What is the value of $f_{1}^{*}(n)$ when $n$ is a positive integer?



Answers

Find a formula for $ f^{(n)}(x) $ if $ f(x) = \ln (x - 1). $

All right, So for this question, we start off with FX is eat with Ellen of X, and then we're asked the question if f of eat the power off and plus one So that is, we'll eat the power of N plus one minus half of e to the power of end equals one. Okay, so all we have to do is we have replied in need, Cooper, that puts one and eat a bar of any intel Innovex we have ln on e to the power of M plus one minus Eleanor to power. And now, using our rules that we know, we know that this simplifies to m plus one, and this simplifies to end. So the ends cancel left That equals one for any value of M. And so we have proved it. So this statement is therefore truth. All right, that's all

Okay, So this problem, we're gonna find a few derivatives of R F of X, equal sign of X function slow. Our first derivative of sign of X is the co sign of X. Our second derivative is the derivative of that which is the negative sign of X. Our third derivative of X, um, is going to be the negative co sign of X and then finally, the fourth derivative of X. Um, we're gonna get the negative negative sign of ex, which is just the positive sign of X. Now, it does ask what would happen if we had, um, the fore end derivative of X. So this is if n is one right here. But if and his two would be the eighth derivative now, you'll notice that the fourth derivative matches the original derivative. So we'd cycle through these same four derivatives. We'd get the fifth. So this is the fourth derivative. This is the fourth derivative. Then this would be the fifth. This would be the sixth. This would be the seventh, and then we're back here for the eighth and we just go through this cycle over and over and over, and every multiple of four, we'd get back to the sign of X. So we can just say that as long as N is, um, a natural number right here after the foreign derivative is just gonna be the sign of X as well.

Love forty four for F X equals Kasai Axe. We want to find of ex primed up a problem trip up Ron and F X Closely related and wass have foreign over Lex where on is a long active interest. You're so I have because cause I axe of X prime equals negative sign X right of x double primate perspective. Because I had has extreme Bill prime Because science Yeah, I have access to the force derivative because because I asks again. So So you see, the patterns is a like period. So I have four on Dax's just cause I ax goes backto side axe for and non activity.

This is the fun problems we have dysfunction FX. It is natural log of X, and we want to take the first derivative second rivet, third route of fourth derivative. Okay, so that's going to be straightforward. But the goal is ultimately, Teo. Guess of formula for the insta riveted. And when you're doing something like this, what we're doing, what we're going to do is we're just going to start taking derivatives and we're looking for a pattern. This is something that comes up a law and calculus, more advanced calculus. And, of course, hire your math. You know, if I want to know what the derivative of the function is, I can either just, you know, one of the hundredth derivative. I could just take a hundred derivatives. Or if I could figure out what the general form of the derivative is, I would just have a formula for the hundredth rivet. So let's just start taking driven. This first derivative is one over eggs. Second derivative. Okay, this is extra demise. Once that's minus one over X squared, then third derivative. Okay, so that's negative. X to the minus second. That becomes too over X to the third, fourth derivative. This is to extend a minus thers. That's negative. Six over X to the fourth. Okay. And you can start to see a pattern here. Maybe we'll even do one more. So this is extra minus forth. So again, bring down that negative for it becomes positive twenty four x to the minds fifth. But then excellent mind Smith conquered the bottom. This is twenty four over X to the and so we see right away a couple of things. First of all, recounting Answer. We have it first of all, so you take the first derivative, we get a positive answer. We take the second derivative we getting negative Answer thirty positive for through Negative. So the sign is alternating. So what that means is we're getting a factor of minus one, but it's either going to be plus or minus one, depending on end. And we know the first derivative is positive. So for you have negative want the n plus one. We see that when innes odd. So one three five, this is going to be an even number in the power of negative one is going to make it positive one. And if it is evil two for and in six, etcetera. We're going to get a negative here because it's going to be negative. One to remember, even plus one would be on. And this is a con trick to kind of showing off alternating sequence like this. And the next thing to notice is that the number in the numerator kind of has a special form. Okay, so this is just one. Okay, I can think about this is being one times one I can think about this is being one times two I can think about this is being one times two times three I can think about this is being one times two times, three times four. Cause if you remember, that's how we're getting. These numbers were taking the previous number and multiplying by whatever derivative we're taking. So these numbers one one times two, one times two is three. One times, two times three times for etcetera. Have a special name. These air the factorial CZ, this is one exclamation mark on factorial misses two exclamation marketed factorial. It's three factorial, spore, factorial and in general have in directorial. And so we see that. Okay, if we take the first derivative. This's actually zero factorial. That may seem a little bit weird, but just by convention zero factorial is equal to one pray and now this is going to be one factorial to factorial three factorial. So we take the end derivative. We're actually getting in minus one sectorial. There's still a little off set and then we see, then the denominator. We have exactly X to the power of n. So here's our formula of the instrument ihe ve a national of X.


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