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Find $d y / d x$ using any method.$$x^{3}+y^{3}=10$$

$$-x^{2} / y^{2}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Missouri State University

Campbell University

Baylor University

University of Michigan - Ann Arbor

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 need to take the derivative of X cubed plus y cubed equals 10. Now we could do this either with explicit or implicit differentiation. But since this is a lesson on implicit differentiation, let's do it that way. That means I don't have to solve it for why I could take the take the derivative off. All of this just all jumbled together, the way it ISS that what I have to remember is when you take the derivative of why you're really using the function, the chain rule, even though you can't see it, because why is a function of X? So if I had something like five y when I take the derivative of this the derivative of five, why would just be five? But why is also a function of X, So we would have to tag on that D Y d. X onto it. So we will do that here. In our particular function, let's take our derivative X cubed becomes three x squared derivative of y Cubed is three y squared times. D Y d x. There's that chain rule and the derivative of 10 is just zero. So I'm going to divide every term by three. So those three, they're going to be gone. Non de y dx terms get moved over to the right hand side. That becomes a negative X squared, and I'll divide by why squared? And that gives me my derivative negative X squared over y squared.

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