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Find $d y / d x$ using any method.$$x^{2}-x y+7=0$$

$$\frac{2 x-y}{x}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Missouri State University

Harvey Mudd College

Baylor 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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this problem we need to take the derivative of X squared minus X Y plus seven equals zero. Now we have two ways We go about this, we could solve this explicitly, which means we're going to solve for y and then take the derivative. Or we can use implicit differentiation where we take the derivative just the way it is with everything all jumbled up. We're going to do implicit differentiation here, mostly because that's what this lesson has been about. But one thing to remember when we do this is any time we take the derivative of of a piece with y in it, we need to add on a d Y d X. That's because of the chain rule. You've got to remember that. Why is a function of X okay? Why equals something in X? Just because we don't know what that is doesn't change the fact that it is. There is a function of X. So when we take the derivative, yes, we take the derivative of why. But why stands for something and when you need to put that d y d x on this show that we're then going to take the derivative of that next piece, that next piece in the chain rule. Okay, so let's see how that works in this case. First, we're going to take the derivative of X squared. Well, there's no why they're so it's exactly what we've done up to this. 0.2 X now product, we have X Times y product rules is in effect. When we take the derivative of why we do, you have to put that d Y d exit. So first times the derivative of the second plus the second times, the derivative of the first, which is just one derivative of 70 as is zero. Okay, so I have to get rid of these parentheses, okay? Everything. That's not a term with D y DX. I'm gonna push to the right hand side, and that becomes why minus two X, and then I'm going to divide both sides by negative X. Okay, so that's my answer. I have d Y d X equals Why minus two x over negative X. I don't really like having that negative in the denominator, So I would probably rewrite this and multiply top and bottom by negative one, and that would give me two x minus y over X. Either one of these is perfectly valid ones, not really better than the other. It's personal preference. How you want to write that final answer?

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