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Show by implicit differentiation that the ellipse$ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1 $at the point $ ( x_o, y_o) $ is$ \frac {x_o x}{a^2} + \frac {y_o y}{b^2} = 1 $

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00:41

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

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.

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.

05:07

Show by implicit different…

08:42

03:38

Show that an equation of t…

In this problem, we are given an equation of an aeroplane where, as to show that, at point x, not 1 equation of the tangent line. Is this given equation all right? We know that equation of the tangent line is of the form y minus y. Not is derivative of this function. It will let that given point times x minus x. Not so we need to find a derivative of this function. Lets differentiate the given equation. We have 2 x over a square plus 2 over v. Squared times y prime is equal to 1. Now, let's plug the given point in we have 2 times x, not over a squared plus 2 times y, not over b squared times y prime is equal to it, while this should be 0 since it's a constant. This is 0 from this time we see that y prime is equal to negative b squared over a square times x, not over y, not all right now, if that is y prime, we can rewrite equation of tangent lie. Is y minus y not is equal to negative b squared over a square times x, not over y, not multiplied by x, ponent xro. Now, let's divide both. Let'S multiply both sides by a square times y spread. This would give us a square y, not times y minus a square y, not squared, is equal to negative b squared x, not times x, plus, b squared times x, not square all right now we can write this 1 as a squared times y, not times y Plus b squared times x, not times x, is equal to a square y in s square plus b, squared x, not squared. Now, let's multiply both sides or, let's divide both sides by 1 over a square times b square b would end up with y, not times y. Over b, squared plus x 9 times x, divided by a square is equal to. Why not spread over b, squared plus x, not squared over a square, now, look at the right hand, side. We have here and look at the original equation. If you put x not and y not into this equation, we would get the right hand side right. So we know that this right hand side is equal to 1. So then we end up with y, not over y of b squared plus x, not times x, over square is equal to 1 point and, as you can see, that is actually the equation of the tangent line.

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