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(a) The curve $ y = x/(1 + x^2) $ is called a serpentine. Find an equation of the tangent line to this curve at the point (3,0,3).(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

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01:11

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Missouri State University

Baylor University

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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it's clear. So when you read here so we're given why is equal to X over one plus x square. We have the 0.3 comma 0.3. This gives us a line of lie minus 0.3, which is equal to M times X minus three and we have to find for em. So we're going to get our derivative, which is equal to our slope. I need to get our derivative. We use our quotient. Rule one plus X square, the derivative of eggs minus X times the derivative of one plus next square all over one plus x square square to get one plus x square terms one minus x times two X All over one plus x square square This becomes equal to one minus X square over one plus x square square You plug in three and we get a slope of point negative 0.8 We substituted into this equation when we get why is equal to 0.54 minus 0.8 bucks for part B. We're gonna draw a graph and it looks like this is three. Well, d'oh five. So our graph, it was like this. And then our tangent line We'll cross like this. This point is three comma 0.3

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