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(a) $f(x)=\left(x^{2}+2\right)\left(x^{4}-7\right) .$ Find $d f.$(b) $y=x^{6}\left(2 x^{4}-3 x+4\right) .$ Find $d y.$

$$\text { (i) } 2 x\left(3 x^{5}+4 x^{2}-7\right) d x$$$$\text { (b) } x^{5}\left(20 x^{4}-21 x+24\right) d x$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

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.

01:46

Let $f(x, y)=4 x^{2}-2 y+7…

02:11

02:16

Find(a) $f^{\prime \pr…

05:48

(a) Find $d$ fif $f(x)=\fr…

05:08

$$\begin{array}{l}…

in this question we have to fact we are given F X is equal to X squared plus two. Applied by X. to the power four minus seven. And we are to find a prime X. Or the differential to differential effort primal fix. Now the first impulse will be to looking at it. We see that is in the form of UV And the 50 impulse will be to use the formula do you? VD X is equal to U DVD X plus V D U D X. Um This the product rule however you can see that if you expand this and use implicit differentiation, it is usually simpler that way. Now we're gonna use implicit differentiation by expanding the packets. So FX is equal to X squared by extra. About four is extra about six X squared minus seven X squared times minus seven. It's got minus seven X squared plus two x. to the power four 14. Now there are no like James. So we are just going to now apply implicit uh differentiation here by the parliament. The six comes down six X to the power of five minus 14 eggs Plus eight X. to the power of three and then -14 becomes zero. So this is what we get. We can simplify it further by factoring out too. And the lowest power here is just X. Then here we X. to the power four minus seven plus for X squared. So this is our uh solution for the for the first part next year too. Mhm To to also find it uh to differentiate why is it called? Two X 2.62 X 2.4 minus three X plus four. And we are able to find dy now, as you can see, it's also in that from U. V. And uh but now we know that it's easier to to expand the blood hasn't solved implicitly. So we are going to expand the brackets, we get to X to the power of 10 minus three x. to the power seven plus four X. to the power of six. Now, do I. D X will be equal to I'm solving the Apollo 20 X To the power of nine minus 21 X. To the power of six Plus 24 X. to the power of hold five. Um We can figure out the lowest power here and multiply both sides by dX. So we have our do I. Being able to hear the lowest boys except for five. Then in the broadcast we have 20 X two, All four minus 21 X plus 20 four. And this would be utx Since john McClane had ideas. This would be Our solution for the 2nd part. Mhm. Mhm.

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