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Find the directional derivative of the function at the given point the direction of the vector f(x, Y, 2) = xy + Yz, (1, 5, 6) , (2, -1, 2)}Dvf{1, 5, 6)...

Question

Find the directional derivative of the function at the given point the direction of the vector f(x, Y, 2) = xy + Yz, (1, 5, 6) , (2, -1, 2)}Dvf{1, 5, 6)

Find the directional derivative of the function at the given point the direction of the vector f(x, Y, 2) = xy + Yz, (1, 5, 6) , (2, -1, 2)} Dvf{1, 5, 6)



Answers

Find the directional derivative of the function at the given point in the direction of the vector $ v $.

$ f(x, y) = \dfrac{x}{x^2 + y^2} $,

$ (1, 2) $,

$ v = \langle 3, 5 \rangle $

For this problem, we are given the function F of x Y Z equals the square root of X, Y. Z. The 0.3 to six and the vector V equals negative one negative 22. And we are asked to find the directional derivative of F in the direction of V. The first thing that we need to do is we need to find the unit vector going in the direction of V. So that will be V divided by its magnitude. So it'll be V over the square root of one squared. So one plus two squared, so plus four plus four. So we know that that would be one plus eight. So route nine down below. So we have that this is going to be V or a 1/3 times are vector v negative one negative two, two. To give us the unit vector in the same direction as V. So, having that, our next step is to calculate the gradient of our function at some point X Y zed. So we'll need to apply the chain rule here. We'll have wise said over to root X Y zed as the first component. Then we'll have you'll have X. Z over to root X Y Z as the second component and we will have X Y over to root X Y zed As our 3rd component. Then we want to evaluate our function or the gradient of our function. That is at the .3-6. So we would have let's see here two times six over going to pause and calculate this off screen. All right. So our gradient at the point should be ah one 3/2, 1/2 then to find the gradient or to find the directional derivative that is find the directional derivative of F. And the direction of you evaluated at the point given 3 to 6 we take the dot product of our gradient with our unit vector. So we'll have one times negative 1/3 or just negative 1/3 plus three times negative two. Or excuse me? Plus 3/2 times negative 2/3. So that would be minus 3/3 or just minus. Actually, I'll leave it as minus 3/3 for now and then we'd have plus two or plus one half times to over three. So that will be plus 1/3. Giving us a result of negative 3/3 or just negative one as our directional derivative.

It's video. We are asked to fi direction or derivative off this function if off x y in the direction off Elector we which is described as a plus for J So in the way to fi the directional devotee, usually we know that in by direction we just the fbi, the eggs and in jail direction Wi Fi f by d y by our convention right now, what happened when we have a waiter that is not purely in one direction? The answer is we still fight us component, and then we add them to get the with the coefficient off his directional. This would be one off his directional widows. So since we is created by one company No, I and four J When we fi direction of liberty in i n j coordinate, we at them in the same way. And and this is depending on where you like where the material come form, you may or may not help this last term one over the size. All correct O V. This is too. Megan, are you need director? Because here we have one to the right end for this is all video V is size ese Ah, Route 17. Many others one to defy the directional. There were teams in we direction to be to be a unit So trying it out to size one, That means whatever we get b divided by one over the size which in this case is squire would 17. So, in our case, that's what then one. So we will have this wedding to be ab. Now let's look at the component. So everybody eggs His pasha will be two X plus six. Why pretty easy and might be Why will be six x minus to Why now? Don't forget be one four times the J component. So it will be 24 off ex miners. It'll fly. We at these two together we have that duty. In the direction would be 26 off ex miners to off. Why right times this size off weather we wish is one over. No times one over the site. All victims you issues. So Harry was 17 and this is the answer. This is the It is called the direction of derivative or function f in the direction we very mouthful. And that is it. Thank you.

In this home, you're given the information shown and asked to provide the directional derivative. Our first step in this is going to be the find the partials of ask with respect to X respecto wind and with respect to see and so to find the partial with respect to X Gordon Tree Y and Z is Constance, if you will. If we do that, we get the following either the why plus zero for the middle term plus z e to the X and then I person with respect to why where we treat X and Z is our Constance is going to be exit to the why plus either the Z plus zero for the last term. And our partial with respect to Z is going to be zero for the first term. Why you to the Z plus eat of X. Okay, now we're going to evaluate all of these at a given point, which is 000 And when we do that, we get you to the zero, which is one plus zero times in zero. So we get just either zero or one. Don't we evaluate the second derivative 000 We get zero Trump Seated zero. Which again? Zero plus either zero, which is one we value with ether derivative 000 The zero times in 00 plus eight of the zero, which is one. And now we're going to write these three partials in the vector where the person suspected X That's what I certainly goes. The pressure expected. Why was where I normally goes? And the partial respect is egos or is it only goes they were going to take a directional vector and make sure has a length of one. And to do this, we're going to take it and divide it by the square root of the some of each of its components squared. So we wanted divided by five squared plus one squared plus negative two square. And when we do this, we get the following Five Squared is 25 plus one. Squared is one, so 25 plus one is 26 plus negative. Two square. That's 26 plus four. That's 30. So we get Father wanted to over 30 and I'm going to distribute the route 30 into each term. And so I get five over Route 30 one over route 30. Negative, too. Over Route 30. I'm going to dock products. This with the director I just found. And so when I do that, I take 111 dot product with five over Route 30 one over Route 30. Negative to over Route 30. And when I do this, I get five over route 30 times more money, just 5/30 plus 1/30 times warm to just one over Route 30 plus negative to over 30 times one which is native to over route 30. And so I get 6/30 minus two of Route 30 and that's going to equal for over a Route 30. Now is our final answer. And then I got this by taking the dot product of the vectors with the derivative and the doctor with the direction of the derivatives.

8th grade. I'm inside the park job. Were given a function. Were given a point and were given a victor. And we're asked to find the directional derivative of this function at this point in the direction of the specter. The function is F. Of X, Y. Z equals X squared Y plus Y squared Z. The point has coordinates +123 And the Vector V has components to -12. I remember one time my dad came to basketball practice, I played in a To find the directional derivative of this function. 1st. You want to find the gradients of this function very shitty tile court. This is the vector whose components of the derivatives of F. So we have to X. Y. Lot of Baltimore gems right now, X squared plus two Y. Z. And why squared he would do like that under the hand, granny shot, fucking embarrassing dude. But he wasn't a fucking. And the gradient of F. At our 0.123 is equal to two times one times two is 41 plus two times four is eight. Actually it's not it sorry it's two times two times three is 12. And why? I swear it is four. And so the gradient is 4 13 four. Now our vector V has magnitude the square root of four plus one plus four Which is three and therefore vector you in the direction of V, which is a unit vector Is 1 3rd times 2 -12. Finally the directional derivative of our function F. At the 0.123 in the direction of a vector V. This is the same as being in the direction of our unit vector U. Is equal to the gradient of F. At 123 dotted with a vector U. This is equal to one third times uh Four times two is eight, 13 times negative, 1 -13 And four times 2 is eight. And this is just equal to one.


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