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(2) Let f (x) = Vzx-T. Use the definition of the derivative t0 find f' () . (6) Given that cos (Inx) + bsin (Inx) where and b are real non-zero constants:[5 ma...

Question

(2) Let f (x) = Vzx-T. Use the definition of the derivative t0 find f' () . (6) Given that cos (Inx) + bsin (Inx) where and b are real non-zero constants:[5 marks]Find the derivativesand(4 marks]Hence or otherwise , show thatdx=+y=0,marks]iii. Show by using Leibniz'$ formula that 4n+2) + (2n+1)xy6+"+ (+V)yl) =0 where yln) denotes the n"h derivalive of y.marks]

(2) Let f (x) = Vzx-T. Use the definition of the derivative t0 find f' () . (6) Given that cos (Inx) + bsin (Inx) where and b are real non-zero constants: [5 marks] Find the derivatives and (4 marks] Hence or otherwise , show that dx= +y=0, marks] iii. Show by using Leibniz'$ formula that 4n+2) + (2n+1)xy6+"+ (+V)yl) =0 where yln) denotes the n"h derivalive of y. marks]



Answers

In Exercises $1-56,$ find the derivatives. Assume that $a$ and $b$ are constants. $$f(x)=6 e^{5 x}+e^{-x^{2}}$$

Don't worry. This is a problem from chapter 14 section for you're going to be doing number two, right? So here we have express DW TT as a function of tea, both using the chain rule and by expressing W in terms of tea different shooting directly with respect to tea and then b evaluate d w t t at the given value of teeth. So we have w equals X squared plus y squared in X equals co sign t plus sign t. Then why equals ko 17 minus sign T 40 equals zero, right? So for these problems, it's very important that you remember the formula. So we have dw de ti which equals the partial derivative of W with respect to X times the derivative of X with respect to tea plus partial droning of w With respect to why, in the derivative of why, with respect to tea, right, it's car, eh? So this is gonna be our chain rule. So we're looking for the partial derivative of W, which we said was X squared plus y squared with respect two eggs times the derivative with respect to t of X, which is co sign t last Scientist E Plus you're of W X squared plus y squared with respect to Why times The derivative of why with respect to tea which was co sign t minus scientist, right? So now we can get into this throated of X squared plus y squared with respect to X is just two ex since why would be considered a constant And that's gonna be times the derivative of coastline people Scientist was respect to teeth. So it's going negative. Sai Inti plus co scientist night And so now we need are driven it with respect to why it's gonna be too. Why times again during a co sign is negative scientist E minus scientist e becomes co sign t night. So give me a second and listen to another page in a small room to work night. So at this stage, we have our ex enter wise in her tea and we just want to put everything in terms of tea. So we had that too. Times X, which we said was cose I Inti plus sign t times negative Sorrenti plus CO sign t. And that's gonna be plus our two and then her. Why? Which was Co sign T minus sign T times her negative sign T minus co scientist. Right? So if we want to, we can go ahead and pull that two out of everything and just kind of forget about it for a little while. And now we need to do a lot of boiling. Right? So we have co sign tee times. Negative scientific. It's gonna give us a negative Koh Sai Inti scientist. Then come sign t. So? So there's inquire is that far So we're just going to do something like that. It's normal. Foiling right Signed t plus co sign squared T wass sign tio co sign T minus signs Square too Narrates. That was all the 1st 1 and you can see that thieves actually went to cancel. And so now we can do our second half That's gonna be negative. Co sign T Sai Inti minus co sign squared T wass sine squared t plus scientist e acidity. Right again, those cancel and we can actually see that. That's positive. That's negative. That's negative. That's positive. So this actually comes out to two time zero, which is just zero right, and that would be the answer for a You're right. You also asked us to determine directly so we can look at G. W. Dante. And we would say that D W t was x squared plus y squared with respect to teeth. Great. So we need to change those exes and wise to what they were before. So we knew that X was co sign T plus I, Inti, that's gonna be squared. Plus, then why was co sign t minus sign T? And that's going to be squared. All right, so once again, we have a lot of foiling two. D'oh! So that's gonna be coastlines. Where? T plus co sign T sign T plus co sign T Sai Inti plus sign Square t. And now for our second half that's gonna be co sign squared T moves minus cose I Inti sign T minus co sign T Sign T plus sign square t. All right. So you can see there's negative and positive and the negative and the positive. And now we can combine like terms. Here you have the derivative with respect to Teo of too close in square T plus two signs warranty. Well, we know that co sign squared to you plus sign. Sporty is a pit staggering identity that is actually equivalent to one. So if we take out the two and combined those coastlines, Gertie, people assigned square T is just one. And we have the derivative with respect to t of two. Well, that's a constant. So the derivative will just be zero. All right. Great way to check yourself on a test, do it one way, and then go ahead and do the other way if you have time. All right, so now it's two part B. We were given that T equals zero. So do you. W D T. 40 equals zero. Well, if it was zero, we don't have anything to plug in, so it's still just a zero Lauren.

We are given the function defined as f. Of U. Is equal to five U squared minus. Uh To E. To the U. And there's a couple of themes going on in this problem. When you're asked to find the derivative, the first theme is that we have a power rule right here And hopefully you've seen enough power rules that you've gotten used to multiplying the exponent by the coefficient five times two is 10. And then you subtract one from the expo two minus one, it's one. But you don't have to write uh you to the first power because it's just you. Uh the difference just stays there. It's part of the problem minus. Um And then we have an exponential function here. And what we learn in this section is the derivative of an exponential is actually equal to itself. So it stays to E. To the U. Power. Um And this is your correct derivative. You have the power rule, the difference rule which is just the same. And then the exponent rule is the same as well.

In this question were given Bill you as a function of x y and zed and X Y and zed or each a function of teeth. So we're asked to express the beauty of the function of tea. The first we're gonna do that by using the trainable. So we have Do you buy d T I got you That is going to be equal to Di bi di t Uh um, and you can use question will personally, I like producto a little more So I am going to actually write that as X times that's apart. Negative one. Why times That's the power of negative one. So this means we're going to have to product cools here. I also let's do a part of school for the first term. That's going to be d x d t times that's apart. Negative one plus x times the by d t of that's a part of negative one. And then we will have the white 80 time sets of harm naked of one plus y times deep I d t. I'm said to the negative one. Okay, so now this is me x No team. Let's write that as sure as one over a sad and then we have waas x times. Um um directive of Zach Too hard, negative one by DT That's going to be negative. Said to the part of negative, too Times these are DT. And then this will be the why Day, tee times one over said this is gonna be why times the negatives that the negative too train will also be desired. So before we, um, continue with this, let's actually write down our the x d t d o y d t and d d t so that we can plug goes in So we have exes Coast Gertie. So that tells us the ex d t must be here Coast t and then train rolls all times Negative scientist, which is negative too. Scientist Coast. So why is equal to science? Great E So that means the wide 80 must be equal to two sighing t chain Wal Coast and he's at DT. Since that is one over tea. You can think of that as teeth apart. Negative one. So these I did. She must be negative. Er team to the negative too. I like to think of that as negative one over, he scraped. So now that we have these valleys, we can plug everything into this equation in terms of tea. S o d x t t. That's negative. Two sighing t coast E one over his head. That is one over tea. So one over one over t that just tea. That's Times Team. Plus, we have X. So that is co squared t times Negative said of hard Negative, too. Okay, so again, that is one over team. So what? The negative power means we flip it back up, so that's gonna be teeth grid, but with a negative sign for insights Naked of cheese grade. And these i d. T. Is negative one over. He's quick class. You I d. T s. That's two sighing t coast t times one over that. So that's just heat. And then Plus, we have our wine, which we know is sine squared t times negative sets of hard negative, too. That's negative t squared. And these i d t. That is negative one over TV script. So now that we have everything in terms of tea, let's cancel things up. So we have a negative T squared times a negative one over K squared, so that gets cancelled. Same thing here. We have a negative truth 90 CO C. Times t. And then we had the exact same thing, but a positive value. So they will cancel. So this names we are left with Coast Square T plus sighing square T So that means this is just equal toe one, because close, gritty plus size, great heat is an identity. So we're also actually asked to do to find you i d t. But instead of using Chang Ryul, we want to express the view in terms of tea first. So let's do it that way. So that tells us W is equal to X over set. So we're going to plug our X, and that's straight. And so we know X is equal to post where t So this is going to be co square t over one over tea, which is really tea time Coast, Grady. And that's plus y over said, which is going to be tee times signs 14 So this we can factor out the tea and it becomes coast grey tea plus science. Great e. So again we have our identity, which is equal to one. So we end up with W is just equal to teeth. So this means w the tea is just gonna be equal to one. So in this case, it was actually a lot easier to do with the second way might not necessarily always be the case. Um, but just for this particular went, this was easier. So now part B were asked to evaluate D W T T at a particular given Valley, and the given value is t equals three. But since D W T. T is a constant, it's equal to one. It really doesn't matter what value they ask us, because the W d. T. Of any value is going to be equal to one.

Have FFX equal six You the power five. It's plus you'd bar minus X squared. Therefore FDS X equals six into the upon the excessive use of power five x plus the upon the eggs. Of the four minus X square. This is Regional six into Power five X. A. Into five plus Youtube or minus six. Squire into minus twice, affects the Sequels. 38 Bar five. It's minus two weeks. Your bar minus X square. This is our Fs. X and this is the answer.


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