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Use implicit differentiation to find an equation of the tangent line to the curve at the given point.$ x^2 + 2xy + 4y^2 = 12, (2, 1) $ (ellipse)

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00:55

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

University of Nottingham

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.

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.

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Use implicit differentiati…

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(a) Use implicit different…

In this problem, we are asked to use efficient differentiation to find the equation of time line of the curve at a given .21 all right. We know that the equation of tangent lines will be on 1 line. An is equal to derivative of this function and will have a given point: multiplied by x, minus x, not where y not is 1 and x. Not is 2 point, so we need to find a derivative in order to do that. Less like derivative on both sides with respect to x, we have 2 x plus for this term, will be using product rule 2 y plus 2 x times y prime plus 8 y times y prime is equal to 0 and, let's pluck given x and y in Order to find derivative at this given point in order to find this term, we have 2 times 2 plus 2 times 1 plus 2 times 2 y prime plus 8 times 1 times y prime is 0 from this. We see that the y prime should be equal to negative 6 over 12 point and that is equal to negative 1 half pint. Now we have everything that we need and we can write equation of tangent line as y minus 1, which is y not times 3 overtime, negative 1, half multiplied by x, minus 2 point from this. We see that equation of the tangent line is that 2 minus x over 2.

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