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(a) Find $d r$ if $r=\frac{20}{\sqrt{5 u^{2}+9}}$,(b) Find $d y$ if $y=x^{5}\left(x^{2}-9\right)^{8}.$

(a) $\frac{-100 u}{\left(5 u^{2}+9\right)^{3 / 2}} d u$(b) $3 x^{4}\left(x^{2}-9\right)^{7}\left(7 x^{2}-15\right) d x$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

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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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in this question we are giving our equal to 20 over the square root of five U squared plus nine. You can see that it's in the form of you over V. And to find the differential of UV. The formula is V D U. I miss you T V. All over we squared now to calculate our D. You. Yeah we are differentiating a constant. We get a zero. Our tv since V is equal to five years squared plus nine to about half tv will be equal to by the polo. They have comes down five years squared plus nine. two of -1 more. Apply by the differentiated value Inside here we get 10 you now this simplifies to half five Or half of 10 which is five. So we already have five you over the route of five U squared plus night. So this is our tv value and now we can use this formula to calculate um the R D U D R do you is given by V. D. You but D U is zero. So we already have zero there subject You DV. Now you is 20. Our DV is given right here which is five. You over the root of five U squared plus nine. Going forward we have er d'you equal to minus 100. You of the root off five U squared plus nine. But this is G R D U. So D R Will be equal to -100. You All over the root off Yeah five years squared plus nine. D You. Yeah and this is our final solution for the face part. No for the second part we are too to find uh we are given the function. Why is he going to extra five X squared minus nine To the Pulp eight. And we are too to solve for do I? Now we can see that this is in the former U. V. And uh D. U V can be found. Dx can be found by the formula you. D. V. D. X plus. See do you T X know how our do you? T X will be equal to five X. to the ball four. Our tv D X will be equal to eight X squared minus nine. Then subtracting Mahan over there. We have seven apply by differential inside we have two X. And this will give us 16 X times X squared minus nine To the power of seven. So here we have our D. V. D. X. And we have our D. U. D. X. Now to find dy dx we you can use this formula You how you is acceptable. five Apply by DVD X. R. DVD exist right here. 16 x X squared minus nine to post seven plus V. D. U plus fee to you however is X squared minus nine To the power of eight right here. Multiply by D. U. D. X. Which is five X to the ball four. So solving for Um simplifying this we have 16 x To the power of five. Applied by X squared minus nine to the post seven plus. Oh here we have another X ray 16 X to the power of six. You have a six right here. 16 extra post six X squared minus 9 to 12 7 plus. Here we have five x. Just both four well played by X squared -9- 12 8. We can see that the X squared minus nine is common, both both on both parts. So we can actually affect that out and we have an extra before here with the next 12 6 years. So taking uh the common factors which would be X 244 and X squared minus nine To the past seven Inside we are going to be left with 16 X squared plus Here we're gonna be left with five and X squared since we left just will have to do one. Gonna put the five inside. So five weeks squared minus 5.9 which is 45. Yeah now this can be simplified further, you know as you can see we can go forward by um adding this up. So we have X square extra before X squared minus 9 to 12 7 16 X squared plus five X squared we have 21 X. Squared Mhm. Mhm. Yeah We have 21 x squared minus 14 flights. We can pull out the common factor here and put it outside which is 21 and 45 3 is a common. Um Come on factor and That would give us three X. Did about four X. Squared -9 To the ball seven and here we are left with seven X. Squared minus 15. And this he's dy dx. So if we just want dy we multiply both sides by D. X. And we're gonna get these D. X. And this these are final solution. Mhm. Mhm.

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