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Harvey Mudd College

University of Nottingham

01:59

Nutan C.

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

01:34

Scott N.

00:51

Heather Z.

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Felicia S.

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All right, So this is a really interesting problem, because what we want to do is actually verify that in for this particular function, it doesn't matter whether we take the derivative with respect to X or why first in the mixed partial derivatives now, he said that most of the time this is going to be the case, but it's super cool to actually see it in practice. So let's just start taking derivatives. So the partial with respect to X this function well, this is going to be a chain rule. So we have to. And then the derivative of square root of something is one over two times the square root of that something. And then we use the general. So we multiply by the derivative of the inside with respect to X, which y squared is just a constant. So we just get one. Or in other words, this is just one over square, rude of X plus y squared, okay. And now the derivative of dysfunction. With respect to why it was very similar. So we have to, and then we're taking the derivative of the square root of this guy. That's to square it of X plus Why square and then now by the chain rule were multiplying by the derivative of the inside. With respect to why so the derivative is just too. Why like that? All right, So in other words, this is just to why over square root of X plus y scream. Okay, so now let's take the derivative with respect to X and take the derivative with respect Toe walk. All right, so we want to take the derivative of this function. Now this function is just X plus y squared to the negative one half. So the derivative is negative. One half x plus y square to the negative three halfs times The derivative of the insight With respect to why which is just too why or in other words, the one half cancels with the two and we have negative why over X plus y squared to the three haps. Now let's take the derivative of f sub y with respect to X. So we're gonna take the derivative of this with respect to X. But notice that this is just too why Times X plus y squared to the negative one half. So the derivative with respect to X. This is just a constant. So we have two y times. We bring down the negative one half and then we have X plus. Why squared to the negative one half and then we multiplied by the the derivative of the inside with respect to X, which is just one. Now, look at this. The one half cancels with the to on that's, uh, three house. All right, bring down the negative one half and subtract one. And what are we left with? Negative. Why? Over X plus y squared to the three halves, which is exactly the same thing as we got when we took the derivative with respect the experts and then whine Amazing.

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