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A panicle moves along a straight line with displacement $ s(t), $ velocity $ v(t), $ and acceleration $ a(t). $ Show that$ a(t) = v(t) \frac {dv}{ds} $Explain the difference between the meanings of the derivatives $ dv/dt $ and $ dv/ds. $

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$v(t) \frac{d v}{d s}$

01:13

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

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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.

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Our goal is to show that this equation is true and also to talk about the difference between DVD tea and DVDs. So whenever you have this kind of derivative notation, think about the units on the top and the units on the bottom. So the V stands for velocity. So is the rate of change of velocity and the T stands for time. So it's with respect to time, whereas the other one you still have the V on the top. So it's the rate of change of velocity, but the S on the bottom means position. So with respect to position now, it works out that if you multiply these DVDs times or if you multiply DVDs times d S d T, it's almost like the DS is cancelled almost like fraction problem. And you end up with DVD T so that will help us when we go back and do the first part of the problem. All right, so the acceleration is a derivative of velocity with respect to type, So acceleration is DVD T. And we just saw that we could write DVD T as DVDs times de s DT So DBT t is DVDs times de s DT. Well, what's de s? DT de s DT is the rate of change of position with respect to time. And we know that as the velocity. Okay, so all we would have to do then is rearranged the order of these terms and we have a of T equals B of tee times DVDs.

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