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Differentiate.

$ y = \frac {t sin t}{1 + t} $

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01:10

Frank Lin

03:00

Clarissa Noh

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 3

Derivatives of Trigonometric Functions

Derivatives

Differentiation

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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$ …

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Differentiate.$y=\frac…

02:32

$$\text { Differentiat…

00:56

Differentiate (with respec…

All right, let's take a look at our problem in green. On top, we have y equals t sine of t over 1 plus t, and our goal is to find the derivative notice below i've written down the product rule and the cushion rule, because we get to do both in this problem. So let's get started so the first derivative y prime is going to be noticed that over all we have a quotient. So this will be my? U- and this will be my boy, so that means we're going to start off and we are going to set up our potient rule. First thing we need is prime, we need the derivative of the top, but guess what the top is itself a product. So we have like an f and a g, so f will be t and g will be sine of t. So, let's start off by doing product rule for that first part, so we're going to do the derivative of t, which is 1 time. Sin o t s 1 times sine of t and then plus a derivative of g, so that's cosine of t times f, which is t, and all of that is times v, so we're just we're working our way to the question. Rulin v is 1 plus t point. So there's a lot to this problem. Okay, then minus can, let's do the prime derivative of 1 plus t is just 0 plus 1, so it's just 1 and then we can write, as is our u in this location, because now we've done b! Prime and u and we didn't forget- that's a minus sign- then we divide by the bottom squared so that one's easy, 1 plus t quantity squared okay. Now we got to distribute this all i mean we are done in the sense that we have computed the derivative, but let's go, have fun and see if we can clean it up a bit okay. So what we're going to do is we're goin to do foil on the first 2, basically that binomial by that binomial, so we're going to do foils we're going to do the first terms multiplied. So that's just sine of t- and this is all going to be worked on top okay, then we're going to do outer terms. So we're going to have plus t sine of t, then we're going to do the inner so plus in put that t out in front t, cosine of t and finally the last. So that will be. That t looks like plus t squared cosine of t, and then we still have to subtract of t sine of t, and that is all over 1 plus t square it so we're hoping that something cancels, because this is a really long, numerator. Okay, let's see, if anything cancels we have got at least 1 thing to cancel, so that's really exciting, so i think ramas there y prime, then will be. It looks like sin at. I do notice that i have 2 terms with cosine, so maybe i'll factor that out plus cosine of t leaving inside t plus t squared and then that is all over 1 plus t quantity squared and i think that's as good as we can probably do to Simplify it all right hope that it helps to review of derivatives. I see a next time.

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