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Let z = f(x,y) where f is a polynomial, and suppose x and y aredifferentiable functions of t. We are given the followinginformation:1. When t = 2, x = 5 and y = 3.2...

Question

Let z = f(x,y) where f is a polynomial, and suppose x and y aredifferentiable functions of t. We are given the followinginformation:1. When t = 2, x = 5 and y = 3.2. When t = 2, dx/dt = 4 and dy/dt = -13. ∇f(5,3) = <4,6> Compute dz/dt at t = 2, if possible, or tell that it is notpossible without having more information available.

Let z = f(x,y) where f is a polynomial, and suppose x and y are differentiable functions of t. We are given the following information: 1. When t = 2, x = 5 and y = 3. 2. When t = 2, dx/dt = 4 and dy/dt = -1 3. ∇f(5,3) = <4,6> Compute dz/dt at t = 2, if possible, or tell that it is not possible without having more information available.



Answers

If $z=f(x, y),$ where $f$ is differentiable, and

$$\begin{aligned} x=g(t) & y=h(t) \\ g(3) &=2 \quad h(3)=7 \\ g^{\prime}(3) &=5 \quad \quad h^{\prime}(3)=-4 \\ f_{x}(2,7) &=6 \quad f_{y}(2,7)=-8 \end{aligned}$$

find $d z / d t$ when $t=3$

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In discussion, We're going to find them a look that does end city where they just represent the partial derivatives that the with respect to us and sanitary present the partial Daliberti wolf t that with respect to t now we're given that that is equal to sign off two x plus y and X is given Toby a Suszkin minus t scare. And why is given Toby as a scare plus two skin. Now we're going to find out the value of status with the help of changeable. And according to team rules said, Does that mean calls that over Carl s is equal to cause it over car lex into Karl X over s plus call that over curled buy into cars that told by over girl s That means now we're going toe Differentiate to that with respect to X And after different shootings that we respect to answer, we get two course off two x plus by and by differentiating x respecto s, we get to us plus differentiating that respect to why we get course off to works, plus by and differentiating why we respect toe us. We get the value off to us. No, by simply keeping it. We have It becomes to four s course off to works, plus by and to us cause off two x plus vice. And by adding we get success goes over to X plus by and by putting the values of X and y in terms off esti we get success because to cause tree as a skill minus p skin. Now we're going to find that t that mean partial derivative Off said, with respect to t applying changeable, we get called that over Girl X into Karl X over called T plus calls that over Carli into Carl by over girl now do financially things that the respecto s we get to off to course off two x plus Why and differentiating x respecto t we get minus duty. Plus, differentiating that the respect to wife we get calls off two x plus y and differentiating via with respect to t we get don't be and under simply thing we get 40 course off two x plus by plus duty course off two x plus vice. Now, after simplifying it, it becomes minus duty cause off two x plus y And after putting the values off except by in terms off SS snt we get minus duty goes off three as a scare minus piece girl.

We have to evaluate, develop service and 30 were that is given to be X into Y minus. Access for two into y and X is equal to s plus t and Y is equal to s minus t. Now we're going to find a belief that is by using chain rule. Now, since we know that that this is the partial derivative you off that respect us so, Zoladex that this is equal to call that over Girl X into Karl X over Karlis. Plus call that over, girl. Why into Carl by over girls now, after different shootings that with respect to X, we get why minus two x y and differentiating excellent respecto s we get one and differentiating that we respect to why we get X minus excess care and differentiating Why with respect to yes, we get one now After simplifying this, we get by minus two x y plus x minus accessible to now After putting the value of x and y in terms off s anti, it becomes s minus D minus two s plus t into s minus T plus s plus T minus s plus t whole skin. Now, after simply find all that we get, it is equal to minus three as a scared plus peace care plus two s. We'll divide with one minus. No, we're going to find them a look that the p that is equal to call that over. Car Lex in tow. Carless x over called T plus. Calls it over. Curled Buy into called by over. Call T No. Why should anyone to have said with respect to X is why minus two x y and differentiating t differentiating X With respect to E, we have one and you can see things that respect to why we have X minus access care and differentiating Why with respect ot, we have minus one after simply think it, we get my minus two x y minus X plus except skill now putting the value of y and X In terms of SD, it becomes as minus T minus two multiplied with s plus tree into as minus p minus as plus d plus as plus cheese, whole skill after simplifying all be I'm just simply saying all that we get C t scare minus as a scare minus Toti multiplied with one minus as that is over. Answer

We have to evaluate very about s N p. So by applying changeable, we get these at over. DS equals two calls it over Car Lex multiplied d x over d s plus calls that over curled y while deployed with d buy over DS Now by different shootings that respecto s X we get two weeks into signed by and differentiating x respecto s we get one plus differentiating that with respect to why we get X scared into cause off by and divide Differentiating why we respect toe s We get zero So it becomes two x into signed by plus zero That but very, pretty well said we destructive t again applying chill rule We get dessert over DT because toe call that over. Culex in tow be X over DT plus vehicles that over girl right into d by. Over DT now putting the value of calls at the worker. Lex, we get two X in tow, signed by and the X over. DT is minus one and girls that over called y is X care into course off fire and dissent cheating. Why, with respect to t, we get do people now The expression is reduces to minus two X sign by bless to excess skin G go. So right now, by putting them in lieu of X and y in terms off variable snt we get minus two multiplied with s minus T in tow. Sign T Scare Plus do de s minus t whole scare into course off piece care. That is our final.


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