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Show that the sum of the $ x- $ and $ y- $ intercepts of any tangent line to the curve $ \sqrt x + \sqrt y = \sqrt c $ is equal to $ c. $

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02:56

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Oregon State University

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

Let $L$ be any tangent lin…

14:38

Tangent Lines and Intercep…

02:55

04:09

Show that the sum of the $…

04:55

03:46

Show that the $x$ -interce…

In this problem, we are given an equation of the curve and we're asked to show that the sum of intercepts of the tangent line to this curve is equal to c. So, let's first find the equation of tangent line. You know that tangent line will be of the form y. Minus 1 is equal to the derivative of y. We elect a given point: minus x, minus x, not less system, not. We are relating that at a point x, 1. All right now we're going to differentiate both sides with respect to x, so we have 1 over 2 square root of x plus 1 over 2 square root of y times y prime is equal to 0 point from the stand. We say that y prime is equal to negative 1 over 2 square root of x, divided by 1 over 2 square root of y, so that is equal to negative square root of y divided by square root of x. Now, at a given point, y prime is equal to negative square root of y, not divided by the square root of x. Not and from this we see that, then the equation of tangent line will be for y minus y. Not is equal to negative square root of y, not divided by square root of x, not times x, minus x, not all right. The female both sides by square root of x. Not we find the equation of the tangent line will be able to form x divided by x, not plus, for root of x, not times why not plus y divided by square root of x, not times y, not plus y, not is equal to 1 point. So from this we see that intercepts or square root of x, not times y, not plus x, not also square root of x, not times y, not plus y, not now. The sum of those 2 would then be x, naught plus 2 square root of x, not times y, not plus y, not, and that is actually equal to sum of x, not plus sum of y, not squared right now. Look at this form so scurf x, not plus for to find out and look at the origin of the question point now. If you were tabulate, this given question at given point x and a cornaro would end up with square root of c, so that will be equal to the square root of c square, which is equal to c. So we just show that the sum of intercept tangent line is equal.

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