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Find $ y" $ by implicit differentiation.$ x^2 + xy + y^2 = 3 $

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01:12

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

05:50

Find $d y / d x$ by implic…

01:52

01:10

04:09

In this problem were given an equation that is x, squared plus y plus y squared is equal to 3 and were as using differentiation to find second derivative of this question. Let'S we find first derivative by taking the veto of old terms with respect x. We have 2 x plus y, we are using for doctor for the second term, on the left hand, side plus x times y prime plus 2 y times y prime is equal to 0 point from this. We see that y prime is the negative of 2 x plus y, divided by x, plus 2 y, all right now using this, let's find second derivative we're going to use quotient rule. So we have negative of 2 plus y prime multiply by x, plus 2 y minus 2 x, plus y power 1 plus 2 y prime divided by x, plus 2 y square y prime, is an equal to negative of now. We know what y prime is so. We'Re going to take this and then plug it in here and then plug that in right, here, 1 of c r prime. So we have 2 minus 2 x plus y divided by x, plus 2 y plus pi by x, plus 2 y minus 2 x, plus y multiplied by 1 minus 2 times x, plus y sorry, 2 x, plus y divided by x, plus 2 y divided by X plus 2 y squared, we can write this 1 as y double prime is equal to negative of 2 times x, plus 2 y minus 2 x, plus y multiplied by 3 times x, plus 2 y minus 2 x, plus y multiplied by x, plus 2 y Minus 2 times 2 x, plus y, divided by x, plus 2 y t now y double prime, is an equal to negative of 3 y times x, plus 2 y minus 2 x plus 4 times negative 3 x, divided by x, plus 2 y cube, and that Is equal to negative 6 times x and 4 plus x, squared plus y squared divided by x plus 2, now look at what we natardin the numerator and look at the original equation given here? Those are the same. So we know that numerator is equal to 3. From this we find y double prime to be that negative 18 divided by x, plus 2 y j.

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