~ulus1f f() =7+%find f' (z). You should show your work for this probler:.Answers (in progress)...

find $f_{x}, f_{y},$ and $f_{z}$. $$f(x, y, z)=x y+y z+x z$$

Hi there in this problem, were asked to find all three partial derivatives of this function. Free to the variables before we start him is going to quickly rewrite this to change that radical into a 1/2 power. We often do this in calculus just to make her shortcuts easier. Okay, so now we're ready. Let's take the partial derivative with respect to X. Remember, we're treating X is the variable. And why? And dizzy both as Constance here. So the first term x has derivative one. The second bit notice doesn't even have an accident. Why? And zero both. Constance. So this is just subtracting off one big constant which will have derivative zero. So we're done with that partial derivative of X now the partial derivative with respect to why this time that first term X has driven a zero with respect to y You know, looking over here how we rewrote things, we keep the mind. A son now treat wise variable Z just as a constant. So we're going to need the chain rule. Well, you can see we have some inter function. Why squared plus c squared. Although the 1/2 power So the chain rule says the derivative of that leaves the power rule on the there we lose the power rule on the outside so that 1/2 comes in front. We keep that inter function exactly as it is just that we wouldn't help. One power drops down by 1/2 and the chain rule tells us we need to multiply by the derivative of this inter function. We just need to be careful, because by derivative, in this case, we mean the partial derivative with respect to why that's the only thing that's different about these problems is that you have to specify which drifted if you're taking the partial with respect to okay, so again, we had that minus 1/2 my squared plus C squared to the minus 1/2. You know, the partial derivative of y squared plus C squared with respect to why is just too. Why, since that Z is a constant and so we can clean this answer up a little bit, there's two, and that to Owen, bottom will cancel out. So we're left with looks like a minus y on top and on bottom, since we have a negative 1/2 power will have a square root of pi squared plus c squared. All right. And finally in the partial with respect to Z the same idea here This will look very similar since, uh why in the Z kind of play symmetric rolls here. So the first term X has derivative zero again, With respect to zem, we still need the chain rule for the second bid justice before So it's still 1/2 times air inter function to the minus 1/2 We still multiplied by the derivative The only difference is now bye derivative We mean the partial with respect to Z because the only difference and so rewriting everything it's only the very last part that will change the derivative The partial derivative of y squared plus C squared with respect to Z is to Z not to our We get the same cancellations as before and her final answer We get minus Z on top all over the square root of y squared plus c squared and we're done

Either in this problem were asked to find all three partial derivatives of this function. So we'll begin with X notice. We have e to some function, so we'll need the chain rule here. So we'll start from the outside. The derivative, as always, eat anything is itself e to that exponents. Now the change will tells us we need to multiply by the derivative that exponents and by derivative we mean the partial derivative with respect to X of that exponents. Okay, Nothing. So we will keep our eat on minus X y z either and the partial derivative of negative X y z with respect to X Remember, we're treating exit are retreating Why in the Z as Constance retreating X like the variable. So the derivative of negative X y z with respect to X is just negative, y z I was using the power rule on exit goes from X to one So that's all they get, So final answer will be negative. Why z e to the minus x y z Okay, now partial with respect to why this will feel exactly the same Once again we start from the outside so hee to that inter function this time he wanted multiplied by the derivative. But it's the partial derivative with respect to why of that inter function as the only thing that's different this time we can see what's going to happen. You should look at this The partial with respect to why now we're holding X and Z Constant. Why is the variable also look at minus X Z for a final answer of minus x z e to the minus X y z All right, this is gonna feel exactly the same for Z in this function X y z or symmetric. So because they all play the same role, this is gonna feel like doing the same thing three times. It's not always like this. So we have e to that inter function this time we multiplied by the partial with respect to Z. Now that you've already seen it before, you know what's gonna happen. This partial derivative with respect to Z is negative x y and so that'll get multiplied by air hard in the beginning here and we're done. Hopefully that

Either When this problem were asked to find all three partial derivatives of this function here, Um, no, I don't know how long it's been since you've done, uh, inverse trick derivatives. So let's just keep in mind in the corner that with one variable the derivative of inverse sine of X is one over the square root of one minus x square. I hear, of course, you have three variables, but this room was gonna be necessary to get the answer. So let's begin with the partial. With respect to X, we're treating wine. Zia's If they're just Constance here. All right, So if we think this has a chain rule, where are the inter function is right here. Why did Time Z Times X almost will start from the outside so the derivative again will be one over square root one minus this whole inter function X Y Z squared. Okay. Now, by the chain rule, we need two multiplied by the derivative of that inter function X y Z, except by derivative. Now you don't just mean the derivative without having to say anything else. We specifically mean the partial derivative with respect to X of that inner function there. So again, this is just the chain rule. Just we have to specify which derivative we're taking which variable we're taking the partial derivative with respect to In this case, it's X Okay, we're almost done on the derivative of X y Z with respect to X is just y z. Since we're treating lines, he is Constance. So we get y Z. We could put that on top and the bottom will remain exactly as we had it and we're done with that partial derivative are the other two you'd expect full of very, very similar since X y and Z Earl symmetric in this function, so just gives us more practice once again start from the outside. The derivative of inverse sine is one over square root one minus that inter function squared This time, the general says. And when we multiply by the derivative of the inter function what it means if you multiply by the partial derivative with respect to why the partial derivative with respect a why we treat wise. The variable X and Z is constant, so that will give us X Z and everything else can stay as it is and we're done with that one. And I'm sure you can predict what this one will end up as well be complete here. So again we start with one over a squared of one minus x y z all squared this time Multiply by the partial with respect to Z of the inter function, which is X y. So get X Y on top over our usual denominator. And there we go. We've got all three. Hopefully that help.

Hello. Director Xavier Group is our problem. But that fight here diffraction effects Goodbye. Desert in tow. Let X squared right cause is it cool? We must do differently with respect, Rex, It was badly they didn't do lacking by take the square by Collins There Indo do pigs, right? Cause that then different. You know that us for Dubai it will get 31 divert But it's a square, right? Calls there into exits. Clad Go sit then have said what we had to do You want me to and let paying for square away calls There bless There didn't do well Im divided but extra square right calls third it go My goodness side that we will get like that is absurd It will be good that little X squared, right? Call it sick Minus said hands that do it excludable