5

I0. By Newtun' $ Mtthed _ Solv € +e eq-~+on * E3tln Hx= f l)= x23+ Ln Yx f '(x) = 2X 4 4x 2x+_ X = | Xz = /.26456&52 4 Xs= (.1973 4441/7 Xy = |....

Question

I0. By Newtun' $ Mtthed _ Solv € +e eq-~+on * E3tln Hx= f l)= x23+ Ln Yx f '(x) = 2X 4 4x 2x+_ X = | Xz = /.26456&52 4 Xs= (.1973 4441/7 Xy = |.197333847

I0. By Newtun' $ Mtthed _ Solv € +e eq-~+on * E3tln Hx= f l)= x23+ Ln Yx f '(x) = 2X 4 4x 2x+_ X = | Xz = /.26456&52 4 Xs= (.1973 4441/7 Xy = |.197333847



Answers

Let $f(v, w, x, y)=2 v^{1 / 2} w^{4} x^{1 / 2} y^{2 / 3} .$ Find $f_{v}(1,-2,4,8)$ $f_{w}(1,-2,4,8), f_{x}(1,-2,4,8),$ and $f_{y}(1,-2,4,8)$

In this question we have to fact we are given F X is equal to X squared plus two. Applied by X. to the power four minus seven. And we are to find a prime X. Or the differential to differential effort primal fix. Now the first impulse will be to looking at it. We see that is in the form of UV And the 50 impulse will be to use the formula do you? VD X is equal to U DVD X plus V D U D X. Um This the product rule however you can see that if you expand this and use implicit differentiation, it is usually simpler that way. Now we're gonna use implicit differentiation by expanding the packets. So FX is equal to X squared by extra. About four is extra about six X squared minus seven X squared times minus seven. It's got minus seven X squared plus two x. to the power four 14. Now there are no like James. So we are just going to now apply implicit uh differentiation here by the parliament. The six comes down six X to the power of five minus 14 eggs Plus eight X. to the power of three and then -14 becomes zero. So this is what we get. We can simplify it further by factoring out too. And the lowest power here is just X. Then here we X. to the power four minus seven plus for X squared. So this is our uh solution for the for the first part next year too. Mhm To to also find it uh to differentiate why is it called? Two X 2.62 X 2.4 minus three X plus four. And we are able to find dy now, as you can see, it's also in that from U. V. And uh but now we know that it's easier to to expand the blood hasn't solved implicitly. So we are going to expand the brackets, we get to X to the power of 10 minus three x. to the power seven plus four X. to the power of six. Now, do I. D X will be equal to I'm solving the Apollo 20 X To the power of nine minus 21 X. To the power of six Plus 24 X. to the power of hold five. Um We can figure out the lowest power here and multiply both sides by dX. So we have our do I. Being able to hear the lowest boys except for five. Then in the broadcast we have 20 X two, All four minus 21 X plus 20 four. And this would be utx Since john McClane had ideas. This would be Our solution for the 2nd part. Mhm. Mhm.

Section $3.6.1 Problems dealing with the chain rules. So I have why, as a function of you and I have you is a function of X. So in order to find d y d X, so do I. D. X is going to be f prime evaluated at g of x times G prime of X So what do we know here? We know that f prime of you take the derivative that is just going to be six and then we know that G Prime of X is going to be four times 1/2 times x cubed. So this is just to execute. So it tells me that to evaluate this derivative F prime evaluated at G of X is six and g prime of X is two x cubed. Therefore, the answer is 12 x cubed.

For the given function we have effectively Porto. They left $5 which in quarto taken by us a constant we'll have held by Dalits. Helen explodes to life that will the Porto Why? And to express toe Because derivative of Ln expecto one expressed a wife similarly as why will be then let my del boy so that can be protected by tell by us. Why Ellen Explosive toy. You have your great excess a constant So that tells the Porto by in total. But Dalva off held them. Explore toy. Plus Ellen explores to boy and to del Valle del y. Why is interpreted code between the court took while took a while and we have to work? Yes, Ellen Experts store and took one. Then it'll be a condom to live. I expressed away. Yes, Ellen, Exploratory.

Given the function F of X y Z equals natural logarithms of the absolute value of X square Miners XY Square plus wide to the fourth Find the three first order partial derivatives off F Respect to X, Y and Z, and then the mixed second order Partial derivative respect to watch and see. So first calculate the personal derivative off F respected eggs at any point. X y z. For that, you can see there or treat eggs and why Sorry y and Z as constants. So we know that derivative off the natural luxury them of the absolute value of expression is equal to one over expression and say the absolute value e squared minus five Eggs e square. Plus I went to the fourth and all that get to be multiplied by the partial derivative of a bit of respect to eggs off the expression inside the absolute value. So he's multiply by more derivative of X square perspective. Eggs, two eggs derivative off negative five x z square Respective exists negative five z square on the partial derivative off what to the fourth respective excess Syria. So this is the derivative, and that is going to you want to play by one at the numerator, so we have two eggs minus five z square, divided by X square, minus five x z Square plus what to the fourth? No, we calculate a partial derivative respect. Why, at any point X y z that's equal to again. You know that this is equal to one over the expression, said the absolute Father. You so is X squared. Finest five eggs C square Bless Why did have plus y to the fourth and all that smoke your blood by partial derivative of the expression inside the absolute value of res respect. Why the 1st 2 terms don't have been a y. So they're the riveted are zero on last term. It's the River Tiber respect. Why is for white to the third this one to blame but one in Reiter So f for what? To the 4th 4 wide to the third story over Ready by X Square, minus five a. C square plus y to the fourth. Okay, and now we're going to have partial derivative off half terrific dizzy that the point X y z again, that's equal to one divided by the expression inside the absolute value. So is square Sorry Square minus five Eggs Z square plus y to the fourth and hold it. Get to be multiplied by partial derivative of the expression said the absolute value with respect to Z So we have partial derivative of X square zero partial derivative off negative five exceeds square respect Z is negative then ecstasy on down the river Tim Perspective Easy off. Why did the fourth is you? So we have this expression here. Well replied by wind the numerator is going to be negative then Eggs Z divided by big square minus five eggs C square Plus I went to the fourth. Okay, No, we get to find the mixed partial derivative respect of y z. So we get to multiply. We got to find the derivative off thesis expression here. Respect to why So we have that The partial derivative off efforts Victor y and Z at any point. It was C sequel to partial derivative. We respect to why off the person derivative respect z at any point. And we know that personal derivative respect to Z was fun before and he's the suspicion here. So we could be that were there. Get if 10 eggs e negative 10 x z Bye bye X square minus five x c square less Why did the fourth Okay? And we see that the quantity in the numerator is a constant so we can get out of the derivative. So is negative. Then x c times the person derivative Respect who I off one divided by X square minus five x z square plus where to the fourth and that can be written as negative power so we can write it like X square minus five x c square plus y to the fourth. Well, that raised to negative one. That's equivalent to one over X squared minus five x c square plus went to the fourth so deserve a tive Seek alternative 10 eggs e times. So we have the derivative of the power. So we have the exponents Negative one which moved to play by the same base x squared minus five x z squared Plus what did the fourth raised to negative one minus one is negative too. Times the person derivative respect to why off the based off the expression X square minus five xy squared plus went to the fourth and that is first term is it's derivative is syrah because it the bend it doesn't have been. And why Same thing for a second term on the last term, it's derivative. Perfect. What is for why? To the third. So finally you find that, um, derivative sequel to Negative 10 times minus wine. Sten. Times four is 40. So you have 40 eggs. Why? To the third Z divided by a square minus five x z squared love. I went to the fourth square, and that's the mix. Partial derivative off if we're just bank the white and see.


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