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Find $d y / d x$ using any method.$$3 x^{2} y-4 y^{2} x+7=2 x^{3} y^{3}$$

$$\frac{6 x^{2} y^{3}+4 y^{2}-6 x y}{3 x^{2}-8 x y-6 x^{3} y^{2}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

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.

06:27

Find $d y / d x$ using any…

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$$\text {Find } D_{x} y$$ …

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04:36

05:01

00:36

Find $d y / d x$.$$

00:51

Find $d y$.$$y=\left(7…

00:33

Find $d y$.$$y=7 x^{3}…

03:06

Find $d y / d x$.$$y=\…

this problem. We have been given a function and we've been asked to find D Y d X. Now we can use any method we want. We could do explicit differentiation where we solve for why and then take the derivative. But looking at this, solving this for why is going to be extremely difficult. So this is a great time to use implicit differentiation. Remember, implicit differentiation allows us to take the derivative when our exes and wise we're all jumbled up. The thing to remember, though, is that why is a function of X even though we're not explicitly solving for why, why is a function of X? So if I'm taking the derivative, for example, I have a Y squared. We'll be looking at in a moment. If I take the derivative of y squared, that's gonna be too Why, But why is not just a variable? Why is a function? So the chain rule says that after I do that outermost piece taking that taking care of the exponents, I then have to take the derivative of the inside piece. So every time we take a derivative of why an implicit differentiation, we're going to tag on that d Y d X because of the chain rule. Okay, so let's go ahead and use that here. Well, the first term is a product. I've got three X squared times. Why? Let's use our product first times the derivative of the second, which is one times d y DX plus the second times the derivative of the first, which is going to be six x. Okay, Next I have another product rule this time because there's a negative there. I'm just gonna pull that minus four out, Um, and just take, uh, the derivative of y squared X again. Product rule. First times the derivative of the second, which is one plus. The second time's a derivative of the first, which is two y de y dx derivative of seven is zero. So that peace is gone. And now I have one more product to x cubed. And why cubed? First times the derivative of the second again, You got that d y DX plus the second times the derivative of the first. Okay, so let's do our best. From this point forward, we're done with the calculus piece. This is all algebra we're gonna combine as much as we can get rid of parentheses and solve for D Y d x. So just cleaning this up a little bit, this will be six x y I've got that negative four to distribute. So minus four y squared minus h x y d Y d x Over here it gives me a six x cubed Why squared d y d x and six x squared y cute hey, in order to solve for D y dx everything with a D Y DX goes on one side. Everything without it goes on the other. So let's make the left hand side be the D Y DX side. I already have two of those terms here. And to get the third I'll subtract six x cubed. Why squared de y dx And over on the right hand side, I already have a six x squared y cube. Sorry, that's a square. Does it look like it? Minus six x y plus four y squared. Hey, now I wanna factor out a d y DX on the right on the left hand side. That leaves me with three x squared minus eight x y minus six x cubed y squared and this side isn't changing. I'm just going to copy it exactly as is. Okay, Last step. Everything in that. In those parentheses on the left hand side. I'm gonna divide both sides by that whole bit. So my numerator is the right hand side. My denominator is what is in the parentheses. It's not a very pretty or elegant, uh, answer. But that is the derivative of are given function.

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