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Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{y x}(x, y)$ for the function defined in the given Exercises.Exercise 2

(a) $36 x^{2}-14 y^{5}+16$(b) $-12 y-140 x^{2} y^{3}$(c) $-70 x y^{4}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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so if you haven't done number two yet, I would go and do that first early to go watch the video because we already found the partial with respect to X and why for these functions. So there's really not usedto go through them again, eh? So if you haven't watched that, I would go watch that first. So if we want to find the second partial with respect to X so let's just come over here and do del by Dell X on each side. So that's going to give us So f with two Xs out there and now we're going to assume this. Why here is a constant. So everything else we would just treat and do the derivative as normal. Eso would be 36 x squared and then, since that is a constant just like 14, we just take the derivative of X, which would just leave us with one. So just be 14 y to the fifth, and then over here, the derivative, that should just be 16. So that is going to be our second partial derivative with respect to X. Now we come over here and do the same thing to get our second partial with respect toe. Why so get f double? Why? And then just this X here we're going to assume is a constant and then everything else, um, or the otherwise we treat these variables so that would be so negative. 12. Why? We use powerful. So we would do powerful here. So why to the four times 35. So that would be minus 1 40 x squared y cute And then the derivative of negative three is just going to be zero eso that ends up being our second partial. Now for the mixed partials, you can take the derivative of either of these either the first partial with respect to X or retrospective. Why? And you should get the same answer, but just kind of double check. I'll go ahead and do both of these just to kind of show how, in this case, it doesn't really matter. Um, So instead of these in your race, so over here, would you del by Dell. Why? So this would give me the mixed partial soap. F y i x um And then So these exes here we treat it as very auras, Constance. Now, so those first two will just be zero so zero minus and then we just take the derivative of why to the fifth. Um, So it would be five times 14 or 70. 30 negative 70 x y to the fourth and then plus syrup. So that just sense of giving us are mixed. Partial being negative 70 x y to the fourth. And we should hopefully get the same thing if we come over here and do the mixed partial with respect to X. So now these wise here we treat his Constance so f x y so that first one is just gonna be zero. Um, and then over here we would take the derivative of X squared. So we multiply that by two would be negative 70 x y to the fourth, and then the derivative of negative three is just zero. And so again, you can see how we get the same thing over here. So again, it doesn't really matter which way you go about doing this. Um, normally, it's just whichever one is most convenient or just looks like it's the easiest derivative to take. Um, there are some cases where you can't, but I would say the majority of the time you could always do this without really thinking about too much

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