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3) a) Find the f(x) derivatives 6x10/3 of the 12x ~2/3 following 41-19 functions 1 {6 pts each}...

Question

3) a) Find the f(x) derivatives 6x10/3 of the 12x ~2/3 following 41-19 functions 1 {6 pts each}

3) a) Find the f(x) derivatives 6x10/3 of the 12x ~2/3 following 41-19 functions 1 {6 pts each}



Answers

Find the second derivative of each of the given functions. $$y=\frac{1}{3}(4 x+1)^{6}$$

In this problem, we need to find the 1st, 2nd and 3rd derivatives of the given function F. Of X is equal to X q minus six X to the power for now let us determine the first derivative that would be the derivative of X cubed minus six X. To the power for. So that would be the derivative of X cubed, which is three x squared minus six times the derivative of extra power four, which is for X. Q. So this will be equal to three X square minus 24 X Q. Next let us determine the second derivative that is the derivative of three X squared minus 24 X cubed. So this will be three times the derivative of X square which is two x minus 24 times the derivative of X cubed which is three x square. So we have six X -72 x square. Next, let us determine the third derivative, that will be the derivative of 66 minus 72 X square. So this will be six times the derivative of X, which is one minus 72 times the derivative of X square, which is two X. So this will be equal to 6 -144 x. Mhm. So that means that the first derivative is three x squared minus 24 X cube. The second derivative is 66 minus 72 X square, and the third derivative is six minus 144 X.

Welcome to enumerate in this given expression we have to find the derivative of the given polynomial. So while we are writing the first given polynomial inside the diabetic operator, we start thinking what we should be doing the next step. So if I simplify this, this would be this. So now the next time the next thing will be term by term differentiation where the multiplied constance will come out of the derivative sign the G D X of extra. The power four multiplied with 13, then minus six. With video itself, X cube minus three into D. D X. Of X plus D D X of a constant 24. So this would be 13 into four in two X four minus one. At this step we will try to write the answer directly minus six into three. In two X three minus one what minus three and two digits of Afghanistan And derivative of a constant is zero 13 into four 52. X cube minus 18 X square minus three. When you get proficient enough, you would be able to come to this step directly from this trip because you will do 13 4 52 correct and X will radios. It takes pulling from 4 to 3. So 52 excuse. So this is how you step to start to do a step jump when you become more advanced. I hope you could understand the solution. Let me know if you have any question.

Given the criminal functions. Let me trench incentive from Mia. Okay on. Get me trenches. A median picker. Okay. And now the first question asking us to find, uh, there s early riff to respect to the banks and opponent minus wanted you. So we had a par minus one. You would be here. So because you should respect to the X. So we need to get in a long about this girl. Yeah, and about this stuff and despond minus want you. We see, then the the function is going down there for this one will be smaller than zero I and then we will do that. See here and then we need to find f number Brand X and apartments want you. So we see along Gove here on the bone minus one to the function Is king gave up here? Give up means then. Secondly, rift. It would be punching dio and then for the B. We need to find out for the partially off the under function. Respected of why? On the par minus one to do so, respecting the wind means that we need to get in a long about. Why? So we have your internal on this line. Now discover. And this girl here and apartments one Joe will say is is going down there for the little, uh, this one will be smaller than zero. And now the f the book Bram. I'm the wine and the permanent wanted to Do you see that the department's wanted you here. The graph in this could give down there far the secondly riff. They must be smarter than zero here.

We want to find the derivative of F. Of X equals one minus six x rays and power of four. At this point in single variable calculus, we should have learned a few shortcuts which make it easier to take derivatives like this without using the traditional definition of the derivative. So let's relate those definitions now they're one through three here where one is the power rule D D. S. X. To the A L A X. To the a minus one to the product rule D D. S. F G equals DFB X. G plus FB GDX and three. The chain rule D. D. S F of G of X is F prime G of X times G prime X. So we can apply the chain rule and the power rule three and one. Just all this easily. We don't need to rewrite our function because it's already in the form that makes it easy to differentiate. So let's differentiate Now F prime is four times one minus six X cube. This is by the power rule. Multiply Enemy multiplied by negative six. Because of the chain rule, the derivative of one minus six X. Inside parentheses negative six. That's what obtained. Final solution. F prime is negative 24 times one minus six execute.


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