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Given that f(z) = 2lh(c) h( - 1) = h' ( - 1) = 7Calculate f' ( -1).Hint: Use the product rule and the power rule_...

Question

Given that f(z) = 2lh(c) h( - 1) = h' ( - 1) = 7Calculate f' ( -1).Hint: Use the product rule and the power rule_

Given that f(z) = 2lh(c) h( - 1) = h' ( - 1) = 7 Calculate f' ( -1). Hint: Use the product rule and the power rule_



Answers

Use formulas (1) and (2) and the power rule to find the derivatives of the following functions. $$f(x)=x^{7}$$

For this one, we have to use the quotient rule and then the chain rule. Before I do that, I'm going to sign the numerator and denominator zero variables here the N function So you are is going to be equal to R squared minus one cute and V of our is going to be equal to to R plus one. What's happening to the fifth? Here we go Now the quotient rule states the derivative of H of our will be equal to the derivative of the top. So you are times the derivative of the bottom of our minus the derivative of the bottom time's top all over the bottom squared. So now to do the derivative of you of our we have to use the chain rule. Mhm. So you prime of r is equal to you Prime g of our times the derivative of G of our so here G f R is going to be equal to R squared minus one. So the derivative a view of G of our will just be equal to the derivative of G f r to the third, because you of our is a function race of third power this can be three times g of our squared and then plug back in. Forgive ourselves. R squared minus one squared. Okay, Now we'll do, um, g of our well, that's just the derivative of R squared minus one with respect to our So that's two ar minus zero or just to our So you crime of R is equal to the C three r squared minus one squared times to our doing the same thing for VFR. So the derivative of the of our is equal to the derivative of the of, um, eight of our times the derivative age of our we're here h of r is going to be equal to VFR, which is to r plus one. So then we prime age of our is equal to the derivative. Uh, h of are raised to the fifth power because each of our was a function race of the fifth five times eight of our to the fourth, calling back in for each of our five times to r plus one to the fourth. Now derivative of age of our is just the derivative to R plus one, which is equal to plus zero or two. So the derivative via bar is just equal to five times to R plus one to the fourth times two. Now we could plug that back into this formula up here, So the derivative of H of R is equal to the derivative of the bottom time's top. So what did we have for that? We have you? Of our was equal to six are times are squared minus one squared times via bar, which was to R plus one to the fifth minus derivative of vfr, which would be 10 times to r plus one to the fourth. And that was times, um, you of our which was R squared minus one cubed all overview of R squared to R plus one squared Vivar was to r plus one of the fifth supposed to 10th, and you can take out a two times to r plus one to the fourth. Yeah, and well, if it's just to our post one of the fourth, you can simplify a little bit so you can get, uh, let's see. Two, two times R squared minus one squared times three are times to R plus one minus five times R squared minus one. All over to R plus one to the sixth. You can simplify that further if you want to. When multiplying out what's in these parentheses here, that's basically the answer.

We have for this function here we have F of Z equals tangent of Z plus call centers. E. That whole thing squared. So again we use the power role for functions which is just really the chain role without defining an intermediate variable. So we pull this down so we get to so the derivative expectancy is two tangent of Z plus coast NFC all to the first power. And then we need to take the derivative of this. So the derivative of the tangent is the seeking squared and derivative of the call sign is the minus sign. So that is our final expression there. The next one we have F of Z equals quantity E. To the Z plus two signs er cube. So caution role for functions. Power goal for functions We pull this off. Shall we get three either the time required to E to the Z plus to sign of Z. And then this becomes A. Two. Then we need to take the derivative of the inside here. And so trivia. Tive of either dizzy is either the Z. And then sign of the Deliver, it was just the co sign. So we get plus two co signers. E. So this is our final expression here. Again. We could probably multiply that out and see if it simplifies, but um it's probably not worth it. Uh The last month, the next one here actually we need to do a little bit of extra business because we need to use the product rule also because we have the product of two things here. Right, so the derivative of this with respect to X becomes this times the derivative of this plus this times the derivative. This shall we get that written out like this? And so now we can use the power goal for functions. And so we bring this down, we get two times this To the first power then. And then the derivative of this winds up being two x. And here we get this bring it down. Power rule for functions. We get a three, then this to the two seconds and then the derivative of the inside times that is six X squared. So just kind of simplifying this a little bit. And again we could expand this all out and see if we can get any further simplifications. But um well I can see, I could have factored one of these guys out and I could have done some factoring here I guess. But we get over here You have four x. Find the quantity two X square X cubed minus three. All cubed times, quantity X squared plus four. That's here. And then we get over here we get let's see 18 x squared times the quantity X squared plus four square times the quantity two X cubed minus one. So let's see here we could have maybe simplified this a little bit more. Let's see here we can at least pull a a two X. Out of there. And then we have two X. Cubed minus one factor in each one and I. X squared plus four in there. And then what are we left with? Um We're left with two um two x cubed -1 squared. And over on the other side we're left with plus nine X. Times I see you x. squared plus four. So again we could have maybe multiply that all out and figure out. But anyway that's um and we factored everything out that's what that looks like. I think I did that. All right. Um Square. Yeah I think so.

A five were given h W. That is a function edge in terms off as a variable. W on. We gotta find that the world differentiation and its value at X equal at W equals one. So let's differentiate both sides So etching Taj W we're gonna use the production because the two functions with two are different functions are products square root the blue. The first time remains Saturday's. The transition of it is to the post. Something is easiest to the post something plus second time masters. Different station off route W will be won over toe route. W All right, Alex, I think, differentiated for the second time right away. So we have, in fact, before differentiating, let's try to rearrange and simplify digitized. So let's take it is to the border blew out. We're left with square W plus one or two times square. W uh, Andi. Okay, now we can differentiate it for the second time. So again the product. So the first thing remains as it is differentiation. Off route. The blue will be one or two route. The blue over here 1/2 is a constant on differentiation off the blue race to the poor minus one or two will be minus one or two times. W race to the poor, minus three over to. So, uh, that is for the first part. And for the second, the second time remains as it is. We differentiate the first one which is just areas to the power of We've got other place, the blue by one. So let's replace the blue by one way, have it is to the part one as e This will be one or two. This will be minus one or four. Plus, this will be one plus one or two on this will again be E s. Oh, this comes out as e over four plus three e or two. If you take the common denominator US four. This comes out as e plus six e. So the final answer is seven year over four

Yeah, so we've got some uh function F of Z over here. We've got the square root and we've had both the numerator and denominator underneath. Uh So we want to find the derivative. So we're looking for f prime of Z. Um So there's several ways that we could approach this and I think the easiest ones to try to use the product rule, which I have written over here to the right, We want to separate this into the product of two functions. Um So we need to do some rearranging first. Um The square root we know is the same as the one half power. Right? And we can also um right, that power in both the numerator and the denominator. So this is going to be the same thing as Z -1 to the 1/2 over Z plus one to the one half. Right, So these are these are equivalent. Um So now I can move this bottom turn up top and make the power negative. So I've got C -1 to the 1/2 times C plus one to the negative one half. And now I have the product of two functions. So I can use this product rule. So let's call this first factor F. And our second factor G. Um So we can see in the product rule we need both derivatives of each of those. Um Each of those factors. So let's first fine F. Prime. And when I say f, I mean this lower case up here, so it's derivative. So um We're gonna do the outside derivative. 1st 1/2 comes outfront And I'm left with the Z -1 To the negative one half. And why? Negative one half? Well I have to subtract one from the expo um And then we have the chain rule which says we need to multiply by the derivative of the inside, But the derivative of Z -1 is just one. So the change will really doesn't do anything here. Um So let's let's rewrite this with positive exponents. So this is the same as one over two Time Z -1 to the positive 1/2. So let's do the same thing with the G. It's going to be very similar. So G prime is going to be equal to Take it out of the outside. So negative 1/2 the plus one To the negative 1/2 -1 is negative 3/2. And again, the chain role says we need to take the derivative of the inside and multiply it. But that derivative is just one. Um so simplifying, we get negative one over to times Z plus one to the positive three halfs. Now we're ready to use our product role and um find the derivative of our original function F O C. So we need F times G prime. So F times G prime. I'm going to have a fraction. So I've got this negative here from the G prime part on top. I'm going to have f. Right, so I'm going to have the C -1 to the one half, and on the bottom, I'm going to have the denominator from G. Prime, so I have to The Plus 1 to the positive 3/2. So this guy right here is F. Times G. Prime. And to this we're going to add G. Times of crime, so give me up a fraction. And on top we have G. Um in fact this G has a negative power, so we probably want to put that on the bottom, so there's nothing on top, so we're just gonna use one as a placeholder. And from the G, we have this Z plus one to the one half is going to be on bottom, so I'm bringing that down to the denominator. And then from the F. Prime, I also have a two And I have a C -1 -1 To the 1/2. So this looks pretty ugly. So let's try to simplify it a little bit. So this guy is G. Times F. Prime. So what's let's try to simplify a little here. Um And the common way to do that is try to get a common denominator. So Our least common denominator between the two. Um We're definitely gonna we're gonna need it too, we're gonna need Z plus one. And what exponents do I need? Well I have one half of three half, so we always take the bigger the exponents. So I need a 3/2s And there's AC -1 over here, so I need that as well. And Uh we want the power one half, so this is going to be my common denominator. Um So let's go ahead. Yeah and write down that denominator. So I'm just gonna copy this guy down. Yeah and now we just need to ask ourselves what needs to go on top here. Well in my first term I am missing this, I Z -1 to the 1/2 on bottom. Oh so I need to multiply by that right? So I've got Z minus one to the one half From the numerator already, I need to multiply by another Z -1 to the one half. So that takes care of the first term and second term, what is it missing? It's got the two, it's got the C -1 of the 1/2. Um It's missing some of the z. Z plus one. So we needed to be three half. So we need to multiply by Z plus one To the one or 2/2ves and then we just had a one on top so that doesn't do anything. We can simplify a little bit more. Try to make this look a little bit nicer. So first off we can combine these guys um So Z -1 to the 1/2 times itself. We could just add those. Uh Exponents together. So these guys This is equal to Z -1 -1 To the first power, right? So we've got C -1 plus Z plus one. And then on the bottom can't really do much there. So Z +123 halfs Z -1 to the 1/2. So we're almost there. Let's go ahead and distribute this negative sign out. So we've got negative Z plus one. It's so it turns out negative Z. And positive C cancel each other out. And Um this one and 1 Gives us a two and the two actually cancels out with the bottom nicely. So we have one over Z plus one to the three halfs Z -1 to the 1/2. And this is going to be our final answer for f primacy.


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