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Find $y^{\prime}$ if $y=\frac{x^{2}}{x^{2}+1}$

$2 x /\left(x^{2}+1\right)^{2}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

Baylor University

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

03:07

Find $ y' $ if $ y $ …

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Find $y^{\prime}$ and $y^{…

04:29

Find $y^{\prime \prime}$

04:37

Find $d^{2} y / d x^{2}.$<…

02:49

Find $y^{\prime}(1)$

01:52

Solve

What we have is another quotient rule here. We can tell that it's a quotient rule because, well, it's a giant cushions. So any time you have a giant quotient, you're going to do a quotient rule. So as I'm looking at this, let's go ahead and to the question rule, just looking at the notation they want like crime. There's a lot of different notation for finding the derivative, which is what we're doing. So what you do is you take the derivative of the top. So the derivative of X squared is two X and you leave X squared plus one on and then minus. Now you take the derivative of the bottom, which is two X, the drift of a constant zero. So no need to write down plus zero on you. Leave the top alone all over the denominator squared. Now, some teachers will let you leave your answer like this, so I'm going to circle it. However, other teachers will say No, no, no. Go ahead and distribute this in eso in the numerator. I'm not getting this with the denominator. You get two x cubed plus two x minus to execute. Well, that's kind of nice things to excuse will cancel. So the best answer is actually simplifying the numerator to be two x over X squared, plus one squared. There you go. That's the best answer.

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