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$f(x, y)=10 e^{x^{2}-y^{2}},$ determine (a) $f(1,1),$ (b) $f(x, x),$ (c) $f(x, y+k)$

(a) 10(b) 10$(\mathrm{c}) 10 e^{x^{2}-y^{2}-2 y k-k^{2}}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 1

Functions of Several Variables

Partial Derivatives

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Lectures

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.

04:14

01:06

$f(x, y)=25 \ln \left(x^{2…

03:46

Let $f(x, y)=e^{x}+\ln (x+…

01:46

Let $f(x, y)=x e^{x+y} .$ …

02:15

$$f(x, y)=-x^{2}+x y-2…

If $f(x, y)=x^{2} y /\left…

01:56

Let $f(x)=\mid x-11$. Then…

01:37

If $f(x, y)=x\left(x^{2}+y…

01:03

All right. Um so we have a function of two variables. Look slightly complicated. It's 10 times e to the export price, Y squared all in the exponents. Um and we want to find its value at these three points and some of them have variables in them. So it's a little bit weird. First ones. Just numbers. That's good. So it's 10 E to the and we're just putting in one for X and one for why and that's really nice. The exponents evaluates to zero, E to the zero is one, So that's just a total of 10. Okay, this is a little bit weirder. We're putting an X for X. So we're going to leave excess alone basically and putting an X for Y. So everywhere I see an X. I put an X everywhere, I see why I also put an X. So this looks a little bit funnier. So X squared minus and I saw wise. So I'm also going to put an X. Um Okay, well it turns out that's really nice as well because x squared minus X squared zero. So let's still eat to the zero. So it's still 10 times one is 10. Um Won't always be the case. This may have worked out to an expression with an X in it. Normally, if you see variables here, probably variables here, not necessarily. Okay, this one almost certainly will have variables in the end. So everywhere seeing exit, put an X. Everybody see why I'm going to put a Y plus K. So this is 10 year to the leave X alone and put a Y plus que for why. And then if you want, you can expand out um that determine the exponent. It may or may not be simpler. You can also, I guess, factor by different squares that also may or may not be simpler. Um It depends on what your instructor is asking for.

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