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Hfx)11X-valuesX-valuesg(2) If f(z) then h(z)f' (2)...

Question

Hfx)11X-valuesX-valuesg(2) If f(z) then h(z)f' (2)

hfx) 1 1 X-values X-values g(2) If f(z) then h(z) f' (2)



Answers

$$\begin{aligned} &\text { If } f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right), \text { find } f_{x}(2,1,-1), f_{y}(2,1,-1), \text { and }\\ &f_{z}(2,1,-1) \end{aligned}$$

Given question. We have to find the the video off differentiation off function I feel which is a close to you upon one plus selling you. So this is in the form off upon J function. So when we apply the caution tool, we get Hefner shoes. It was toe one plus Ellen us square one plus Ellen, you differentiation off you. Yes, minus you and differentiation off. Want less than you like this came from the caution tool. So when we simplify this, we get one place selling you. Minus you. Jeez. Oh, plus one upon you. Divided by one less log us square, which is a questo log you upon one place. Ellen, you square. This is the final differentiation off the given function, I feel

In this question were given a function f on the x y and Z He go to the X Square, man a square square, press the square and when you find a value of the function when acting U minus one, why you go to to and it's equal to three. So all we need to do we we have a X y and Z here, so we get a minus one square minus two squared, plus three square. Then we get any culture one minus four last night. So we get equal to the terminus. Funny, could just six the next one. Which computer? With the value of the function when actually 2 to 1 could u minus 10 to 3. So in the X y and Z here looking and got it to square minus minus one square plus three square. So getting close to the farm on this one plus nine, then we get equal to the three plus ni equal to drill. So that's gonna be the answer for the 1st and 2nd part

So suppose. And given the function F. Of U. Equals Elena View over one plus Ellen of to you. And I want to find the first derivative of this function. Well what's the derivative of just a basic Ln of X function. Well the derivative is just one over what's inside of Ellen. Mx Okay. So one of her exes the garrote Vil innovex. What I see with with the function F is that I have two different functions. I have my top function in my body function. So if I'm finding the derivative of two functions that I'm dividing which rules are going to use and use the quotient rule. So you take the derivative of the top function. Elena view is just one over you. Times the bottom function. One plus Ellen of to you minus the top function Ellen of you times the derivative the bottom derivative of one is just zero derivative of Ln of two. U. One over what's inside times the derivative of the inside derivative. To use just two. I'm doing chain rule here all over My denominator one plus Ellen of to you raise the second power. Okay let's distribute and see how much we could simplify. So if I were to simplified to and to cancel if I distribute one over you this would be one over. U. Plus Ellen of to you divided by you and note that these use don't cancel. Right? Because this used in the Ellen and the she was just not inside the Ellen. They do not cancel. And then this will be Ellen of you All divided by one. You again these use do not cancel All divided by one plus Ellen of to you. This quantity squared now since each term is divided by you, what I could do is I could just move this you to multiply by you to the denominator and then the numerator. I'll just get one plus Ellen of to you minus Ln of you. All divided by U. Times one plus Ellen of to you to the second power. And this will be f. Prime of you. And another way to think about you go into the denominator. What I could do is simply multiplied by you to the top bottom. To use in the top will cancel since they each have you into you in the bottom will stay

So in this question we will understand the concept of the general and how to find the derivative of the U. N. Function. Right? So we have to find F-1. Before that we have to find the derivative of the function effects. So if the sex is a close today by the X. Of fx cost gx they so first we'll apply the product rule. Or we can write this as fx day by the acts of costs TX plus because TX the idea itself effects. So this way simplified with the death of protocol. Right now to find a differentiation, we have to apply the chain rule. So this would be equal to fx dot minus sign. Gx got to dash X plus because TX not have sex. They so this can be written as minus effects J dash X saying G X plus. FDA checks because TX like So this is a question number one Now since we have to find a freshman so substitute the value of access one. So we get -F one. Judicial one signs even plus have this one because they want laid. We have all the values given in the question. We can substitute these values So we get -2, multiplied by one saying judo. Yeah. Plus If this one is -1 cause joe like So this would be equal to -1. So the value of this one is -1. Like.


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