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If $y^{3}=2 x^{2},$ determine $d x / d t$ when $x=2$ and $d y / d t=4$.

$$6$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

Campbell University

Harvey Mudd College

Idaho State University

Boston College

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've been given the equation why cubed equals two X squared. And our goal is to find the value of DX DT With some certain conditions Met X is gonna equal to and d y d t equals four. So two observations I want to make before we start solving this. First, you can see that both of these derivatives are found with respect to t. So when we find the derivative X, we're going to need to add on a dx DT when we find a derivative with why we have to attack on a d y d t. This is the chain rule because we're saying that both x and y are some function off t often in related rates problems. T is time, but for right now, we're just putting it in terms of another variable t. The second thing I want to point out is that there's often a question of when do we substitute like I know that I'm gonna want X equal to. Does that mean I substitute right now? No. First we take the derivative, and only after we've taken the derivative do we substitute in the values that we know we want to find the derivative of the function or the equation exactly as it's been given. Then we can plug things in. So let's do that. Let's find the derivative. The left hand side gives me three y squared and I remember Chain Rule says I have to add on a d y d t or I'm not adding multiplying by D Y d t over on the right hand side. That gives me four x two the first times dx DT. Okay, so now that I've taken their derivative, let's substitute in our values, I want DX DT. So that's going to stay put everything else I should be able to substitute for. So we have a small problem and I'm gonna come back to this one in a minute. I'm gonna leave a blank here for the why? Because we weren't given why. So we have to figure out what to do with that. But let's plug in everything else. D y d t is four access to So everything's been plugged in except our why. So how do we find why? Well, we have to go back to our equation and I'm gonna write this and blue right here. Why cubed equals two x squared. We have X, and there's a relationship between X and Y. So at the point when X is too, I can use this equation to find the value of why. So why cubed equals? Well, access to that becomes eight, Which means why equals two. So if X is, too, why is too as well And I can plug that in to my equation here. So that's going to give me, uh, listen, I want to solve for DX DT. So I'm gonna divide both sides by eight. That's going to cancel. That's going to give me a value of DX DT. Once I simplify everything down, it's going to give me a value of six.

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