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The displacement of a particle on a vibrating string is given by the equation $ s(t) = 10 + \frac {1}{4} \sin (10 \pi t) $ where $ s $ is measured in centimeters and $ t $ in seconds. Find the velocity of the particle after $ t $ seconds.

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00:30

Frank Lin

01:09

Heather Zimmers

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Harvey Mudd College

Idaho State University

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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The displacement of a part…

01:34

05:02

$\bullet\bullet$ The equat…

06:36

The motion of a vibrating …

02:16

A particle executes simple…

03:32

The acceleration of a part…

01:25

A particle moves along a s…

06:27

02:14

01:33

Velocity and Acceleration …

All right, since we are given S of T uh is 10 plus a 1/4 signed of 10 pi t. The theme in this problem is when they ask for velocity, that's the same thing as the derivative of that. Uh and just a reminder that the derivative of a constant is zero. I'll go and write that for right now. And then what we have going on here is the chain rule where the first thing that you do is you take the derivative of the outer function. So the derivative of sine would be the outer function is co sign. And what you do first is and leave the inner function alone. But I have to multiply by the derivative of the inner function. And because T is your independent variable, It's just a constant times t. So the derivative of that would be 10 pie. So when you go to clean up your answer, You know, 10/4 reduces to 5/2. You can stick that pie in the new numerator. Tell you multiply fractions Of 10 PT. Co sign of temp it and that's your answer for velocity and we're done at a unit to it are good.

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