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If $x^{2}+y^{2}=169,$ find $d y / d t$ when $x=5$ and $d x / d t=-6$

$$\pm 5 / 2$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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for this problem, we have an equation X squared plus y squared equals 169. And the goal of this problem is to find d Y d t at a certain instance when excess five and DX DT equals negative six. So a couple of things to point out before we go to solve this First you can see that we're taking the derivative with respect to a new variable t. So what that's telling us is that both y and X are functions of tea. So when we take the derivative of why we have to tack on that d Y d t because of the chain rule taking the derivative not only of why but of what y equals, which is, ah, function of t same thing with X. When we take the derivative of X, we have to multiply it by DX DT as an application of the chain role. Thea other thing I want to point out is we've been given a value for X. When do we substitute that? Do we do it now? Do I have to wait? Well, the answer is we wait. First we take the derivative we want the derivative of the general equation. I don't want to put any specific instances. Is on instances in I want the derivative of the whole equation on Lee after we've taken the derivative, Do we substitute? Because once I have the general derivative that I substitute my values to find what it equals at a certain point. But we have to take the derivative first. Okay, so let's go to our equation and take the derivative Well, x square that becomes two x times dx DT. There's that chain rule we're just talking about and the same thing here, too. Why times d y d t and the derivative of a constant is just zero. Okay, First things first, I'm gonna divide both sides by two. It just gets rid of those coefficients. Makes my number's a little smaller. Now let's substitute in. I want d Y d t everything else I should know. So excess five dx DT is negative six. Now we have an issue I don't have wise. I'm just gonna leave that blank for right now. D Y d t is what we want equaling zero. Okay, I'm substituting for everything except the piece. I'm trying to find. So what do I do about why? Well, I don't know why, but I know how why and X are related. I'm given that in my initial equation so I can go to that equation X squared. Plus y squared equals 169. At this particular instance. I know excess five. So I can plug that in and solve for the why that corresponds with this particular instance of X. So I have y squared equals 144. So why is going to either equal plus or minus 12? It could be either one. So while I plug this in, I'm just gonna put a plus or minus 12 here, and we'll see how what happens with that in just a moment. Okay, let's solve for D Y d t. I have a negative 30. I'm gonna move over, so that becomes a positive 30. And I have a plus or minus 12. D Y d t okay to solve for d Y d t. I'm going to divide by this plus or minus 12. Now what I simplify, this is going to give me five halves and is either going to be Plus, if I had a positive why or minus? If I had a negative, why so two possible answers, either plus or minus five.

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