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If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$
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Masoumeh Amirshekari
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so absolutely. The biggest take away from this partial derivative topic is how to take partial derivatives. So we talked about sort of what they are. They're kind of the slope of the tangent lines of these cross sections of the function. But most importantly, right now, we just want to get used to taking partial derivatives in the method taking partial derivatives in The key is, if you're taking the partial derivative with respect to X, as we're doing first, you just want to treat why is a constant so f some x of x y and notice This is the same thing as what we're looking for. Partial F partial X. I'm gonna differentiate the function with respect to X treating Why, as if it was just a number like two or three or 10 or whatever. So we have a chain role, so it's gonna be too types x y plus one squared and then I want to differentiate, not squared. So it was squared. I bring down the to announce to the first and then I multiplied by the partial derivative of the inside with respect to X, which is well, this is X times a constant. So the derivative of a constant Times X is just equal to that constant. So I can write this as to why Times X y plus one. Now the partial derivative with respect to why is exactly the same thing except I want to treat. Why are X is a constant I'm gonna treat X is if it was just any old number and so I do the same thing. I have to bring down the two because this is a chain role and then I leave the inside alone and then I multiplied by the derivative of the inside which in this case, is just Well, this is constant times. Why so the derivative is just the constant that I'm multiplying. Why by so in other words, this is to x times x y Klitzman and these were the two partial drift. It is with respect to X and with with respect to why of this function. All right, so we're having so much fun. Let's do it again. Okay, so let's take the partial derivative of this function with respect to X. So again, I'm just going to treat why, as if it's a constant. So why squared is just going to be a constant. So all I'm taking the derivative of is this e to the X squared. So that's going to be e so e to the X squared times, the derivative of X squared, which is two X and in times why squared like that and notice I'm leaving. The constant alone is if it's just multiplying by a function of X I'm treating This is a function of X, So I just differentiate the only thing that has to do with X, which is either the X squared so you can write this is do X y squared E to the X squared. I want to kind of clean it up a little bit and now the partial derivative with respect to why I just wanna leave that e to the X squared of them. Because now that's a constant, because I'm fixing X. I'm treating X is if it were a constant. So what do I have? I just hav e to the X square that constant times the derivative of y squared, which is to why so there is my partial derivative with response with respect to why here is my partial derivative with respect to X
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