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(1 point)) Let F(z) = [" V2+3dt FindF(1) =(D) F"(1) =F"(1) =...

Question

(1 point)) Let F(z) = [" V2+3dt FindF(1) =(D) F"(1) =F"(1) =

(1 point) ) Let F(z) = [" V2+3dt Find F(1) = (D) F"(1) = F"(1) =



Answers

Find $D_{\mathfrak{u}} f$ at $P$. $$ \begin{array}{l}{f(x, y)=\ln \left(1+x^{2}+y\right) ; P(0,0)} \\ {\mathbf{u}=-\frac{1}{\sqrt{10}} \mathbf{i}-\frac{3}{\sqrt{10}} \mathbf{j}}\end{array} $$

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.

Okay, when you want to find the linear ization, you can always remember this, the derivative, the the derivative is the slope of the tangent and it's almost the same as F of x minus F of a over x minus A. So cross multiply in front of a times x minus. They equals almost ffx minus F. So if you want to find fx, you can just take off of a plus F prime of a X minus they as long as the A is near the extra looking for. Okay, that's the linear ization. So you're saying the linear ization is almost the same as ffx as long as X is near a. Your function here is D natural log effects. So f of a equals FF Juan, which is the natural log of Juan, which is zero and then F prime of X. That's one over X. And so a primate A is a prime at one, Which is 1/1, which is one. So then the linear ization is F. Today, which is zero, right? Plus F Primary Day one times x -A. So X -1.

Mhm. We want to find the first partial derivatives of the given function of X. Y. Z equals X over Y Z. At the 0.1 negative one negative one. This question is challenging understanding of differentiation of multifarious functions to solve. We need to understand that proportions right. Is we use single variable differentiation techniques differentiating with respect to the differentiating variable and treating other variables for the partial river is constant. So with this in mind we should have everything we need to solve using the example here as a showcase of how to use the power rule which comes in handy for our derivatives. So have access. Fy and FZ are the three directors. We need to find the first derivative of this function By the power will affect is one of the YZ. Considerate of Act as one. The F Y is negative X over Y squared Z squared. Because one of the UAE has a lot of negative one over wide square and similarly Fc is negative X over Y Z squared. That's evaluating our point. We have excellent. One negative one, negative one is one over negative one times negative one. It was one. Similarly, Fy NFC are both 12

Alright. If I were asked to do this problem, I would actually figure out what GF Texas, which might be different than the way your teacher asked you. Well, if you look at the F function integrated as 1/1 plus the square root of something, Well, that's something that we're taking The square root of now is G of X. Well, what is G of X? It's defined as one of her ex. So the big thing in this problem is that you have the So as we're doing the derivative, the big thing is thick ocean. Um, or if you don't like that, you could rewrite this as one plus, um, like trying to think of how to explain this, um, that it makes perfect sense that you could do the square root of X one over the square root of X, which is just one over Route X, which is the same as X to the name one half power. I was just going to write that, but I don't want you toe like freak out on me, um, to the negative first power, because it's in the denominator. This is to the first power in the denominator eso. Now that might make the derivative a little bit easier. And you can see the chain rule where you bring the negative one in front one plus x to the name one half power stays the same now to the negative second power times the derivative of the inside, which should be another negative one half X to the negative three House power. So depending on if your teacher lets you leave your answer like this, I mean, if your teacher let you get away with that, then go ahead and use that otherwise you might rewrite as a negative times. A negative is a positive. The fact that we have one plus X or that's still the the one over square root of X to the second power I just moved to in front. This, too is is what I'm talking about, um, and then the square root of X to the third power, but also being the denominator eso This might be how your teacher wants you to write it


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