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(a) Find $d$ fif $f(x)=\frac{2 x+3}{x^{2}-2}$,(b) Find $d y$ if $y=\frac{x^{2}}{x^{2}+1}$.

(a) $\frac{-2\left(x^{2}+3 x+2\right)}{\left(x^{2}-2\right)^{2}} d x$(b) $\frac{2 x}{\left(x^{2}+1\right)^{2}} d x$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Campbell University

Oregon State 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.

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.

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Find(a) $f^{\prime \pr…

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(a) $f(x)=\left(x^{2}+2\ri…

04:46

(a) Find $d y$ if $y=\frac…

07:02

Find $d^{2} y / d x^{2}$

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Find $d^{2} y / d x^{2}.$<…

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$$\begin{array}{l}…

in this problem we are given Y. Is equal to two X plus three over x squared minus two. And as to find do I? And we can see that this is in the form you over V. So we use the formula to you Over V will be called D. X. Will be equal to if E. T. U. T x minus you. Tv T. X. Hope is over free squared Now we can see clearly that we need to utx and D. V. D. X. So do you T. X. Is equal to. Well we are now at differentiating two X plus three. We get to tv T. X. Yeah Differentiating the Denominator. Get two x. Substituting these into our formula when we have to you overpay Which is the same as do I? How would the eggs? He's The Co two. The formula V. R. V. Is X squared minus two. The U. D. X. Is to subtract you which is um the numerator two X plus three. My play by tv T X. Which is right here. Just two weeks. All this is divided by X squared minus two squared. Next we're going to simplify the numerator here we have two X squared minus four -4 x squared minus six X. All this over X squared minus two to the call to now um If we subject two X squared minus four X. We're gonna get minus two. X. Scared -6. X -4. Okay minus four. Over X squared minus two 2012 2. Um Further simplifying this Taking out -2. We have -2 X squared minus three X minus two. Cool All over? Mhm X squared minus two to the pole. Two is um this is the this is dy dx. So if we just want dy dx comes and multiplies this side to make it this is our final solution. Yeah. Now next we want to to complete the the heartbeat. We have white as our Y. Is equal to X squared what? X squared plus one as as our function. And we have to find the where to find dy as well. You and we just like in the last hour question and do I. D X will be equal to V. Do you T X minus you? Tv T X. All over. He squared. Mhm. Now calculating the U. D. X. We get two X tv T X. We get two X substituting into our equation. Do I? T X is equal to V. R V. Is X squared plus one tens the U. D X. With these two X minus U. Or U. Is X squared. My applied by TV TX. Which is two x. All this divided by X squared plus one squared. Next you're going to simplify them the numerator here. You get to eggs cubed plus two X minus two X. Cute. So this actually cancel out divided by X squared plus one squared. Here are the final uh answer will be two X over two X. two x over x squared plus one pull this squared but this is july T x. So to get do I. Mhm. The metal be do I is go to two x of X squared plus one squared T X. And this is our final solution for part B. Mhm.

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