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01:59

Nutan C.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

00:51

Heather Z.

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Felicia S.

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Jeffery W.

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All right. So this is a rather involved problem. We have to intermediate variables x and Y a dependent variable Z, which is just X over y in the two independent variables S and P. Okay, so let's just go ahead and write down what some of these things are. It's a parcel Z Partial X is just one over. Why? Partial Z Partial. Why his negative X over y squared. Okay, What else? Partial. It's sometimes easy to make the partials look like twos. Partial X partial s is just square root of team partial X partial t is s over to rooty here. I actually mean Thio partial. Why? Partial s is t over to rude s and then partial. Why partial t is equal to us. Okay, so there's all the possible partial derivatives that we're gonna need six and all. So let's look here partial Z partial s recall is we just kind of take the combinations we need, so we need toe. Look at first the intermediate variable X, and then we multiply by partial X partial s and then plus, we look at the next intermediate variable. Why? We're looking for the derivative with respect to this independent variable Ehsh Okay, And now let's just fill in what we know. So partial Z Partial X is one over. Why? Partial X partial s is square it of tea and then partial z partial. Why is negative X over? Why squared times partial? Why? Partial s is t over to right s Okay, so let's actually fill in. What x and y are. So why is t rude s We're gonna have writ t over t route s plus Okay. Negative. I guess I could just rate minus x is s route T. And then why squared is t squared s times t over t route? Yes. Okay. So what do we have here? Well, this is going to be one over. I have left of the square root of TM bottoms, square root of ts and then minus s cancels one of the T cancels. Another square root of T cancels minus 1/2 square. It of ts, which is actually just equal to one over to square it a t times s All right. So now let's look a partial Z partial T. Okay, So what is it? Just by definition, when we look at the two intermediate variables Partial Z, partial X times, partial X partial team plus the other intermediate variable Partial Z partial. Why partial? Why? Partial t Okay, now what is that equal? Just plug in what everything is. So you have partial Z partial X which is just one over. Why? And then partial X partial t is Route t and then partials You partial? Why, That's minus X over Y squared times partial y partial t is hold on a second. Partial x partial t is up here. That's not correct. Yes, So partial x partial t is s over to tea and then partial y partial t is just root s. So let's plug back in when x and y are. So this is s over. Why is t rude s just gonna be too t Yes and then routine coming from here? Yes. So why is this? This is why And then we have a two rooty and then plus or minus X is going to be s Retty times Route s and then all over. Why squared again? Which why squared is equal to t squared s. Okay, so then what do we have we have here? We're gonna have a rude us on top, so square root of us, divided by to t routine and then minus here. The SS will cancel their We'll have a see. That's just gonna be, um what? We can just leave. This is ret T s all over T Square. Not now. That's good enough. So there is the partial Z Partial s right here and then partial Z partial t is right here.

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