1. Find f'(x) if f(x) = √(x + (4)/(x)) + 2 [1] 2. Find f'(x)...

1. Find f'(x) if f(x) = \sqrt{x + \frac{4}{x}} + 2 [1] 2. Find f'(x) If f(x) = 2(1 - x)^4 [1] 3. Find f'(-1) if f(x) = ax + \frac{b}{x} [1] 4. Find f'(1) if f(x) = \frac{x + 1}{\sqrt{x + 1}} [2] 5. Find f'(1) if f(x) = (\frac{1}{x} + 1)(\sqrt{x} - 1) [2] 6. Find f''(1) if f(x) = (3x^2 - 5x)^2 [2] 7. Determine the coordinates of the point on the graph of f(x) = 3x^2 - 7x + 4 where the tangent line is parallel to the line 5x + y = 3. [4] 8. Determine the equation of the tangent to f(x) = \frac{x + 2}{x} at x = 2. [3]

Transcript

00:01 In this question, in the first part we have been given a function, f of x, which is equal to 6 cube root of x square minus 20 times the fifth root of x raised to 4 plus 7 divided by the 7th root of x raised to 5 plus 4 divided by the 7th root of x raised to 5 plus 4 divided by the 6th root of x.

00:30 And here we are supposed to find the fourth derivative of f so let us see how can we do this so first of all we are this f of x so my f dash of x it will be comes out to be 4x raise to minus 1 divided by 3 minus 16 x raise to minus 1 divided by 5 minus 5 x raise to minus 5 x raise to minus 5 x raised to minus 12 divided by 7 minus 2 divided by 3 x raised 2 minus 7 by 6 if we differentiate it again we are going to have the second derivative so second derivative comes out to be minus 4 divided by 3 x raised 2 minus 1 by 3 minus 1 which will be minus 4 divided by 3 only correct then we get plus 16 divided by 5 x raised 2 minus 6 divided by 5 plus 60 divided by 7 x 2 minus 19 divided by 7 plus 7 divided by 9 x raised 2 minus 13 divided by 6 so this is the second derivative now we move to the third derivative so what we are getting the third derivative 16 divided by 9 x raised 2 minus 7 by 3 minus 96 divided by 25 x raised 2 minus 11 by 5 minus 1140 divided by 49 okay and then we get x 6 raised 2 minus 26 divided by 7 and here i will get minus 91 divided by 59, sorry, 54 x raised 2 minus 19 divided by 6.

02:41 And finally, if i differentiate it again, the fourth derivative comes out to be equal to 1056 divided by 125 correct 1205.

02:55 In the denominator i will write x -res -to -16 divided by 5.

03:05 So 16 divided by 5.

03:08 Then i will get minus 112 divided by 27, x -res to 10 divided by 3.

03:16 Okay x raised to 10 divided by 3 and lastly i will get 1 429 1429 divided by 324 x raised 2 25 by 6 25 by 6 plus 2 4 6 here we are having sorry 29 6 4 6 4 6 4 0 divided by 343 and x raised to 33 divided by 2.

03:56 So this is the required fourth derivative that we have to find here.

04:03 Correct.

04:05 So this is easy, but it is a little bit involving some calculation that we have to do.

04:12 Now let's move to the next part.

04:14 In the next part, we have been given x cube minus xy plus y square equals to three.

04:23 We have to find out the second derivative.

04:26 Sorry, the third derivative we have to find out.

04:30 So first of all, let's find out the first derivative.

04:34 So we have to differentiate it.

04:36 So here, what i'm going to get? i will get 3x square minus.

04:43 Here you have to apply the product rule y plus x dy divided by d x plus 2y divided by d x equals to 3 so rearranging will give you dy divided by d x equals to 2x minus y divided by x minus 2y so this is my first derivative correct this is what i am getting is my first derivative and this can also be written as if i split it 2x divided by x minus 2y minus y divided by x minus 2y so this is my first derivative we have to reach to the third derivative so this is the first i can denote it by y -dash as well so i now have to find y double dash so i will apply the cosine rule here so that will be x minus 2y whole square, then the denominator here, derivative of numerator minus numerator times derivative of denominator.

05:51 It will be 1 minus 2, d .y divided by d x, that is, y, dash, minus.

05:58 Here i will get x minus 2y whole square, then denominator times derivative of numerator.

06:06 So it will be just y -daz minus numerator times derivative of denominator.

06:12 1 minus 2 y dash this is what i get so solving this if i simplify it okay i am going to get six times of x squared plus y square minus xy divided by x minus x minus 2y whole cube okay so this is coming out to be six times of three divided by x minus 2y whole cube why i'm saying this because it is already given x square plus y square minus xy is equals to three that is still is given in the question so this is what i'm getting so this is my y double dash 18 divided by x minus 2y whole cube now i have to this is what we have to find it out so this is the required second derivative of y now let's move to the the next part part three so in the part three what we have been asked we have to find out the equation of the tangent equation of the tangent to the graph whose equation is y is equals to x root x minus one at five comma ten so let us see how can we go about it so first of all we have to differentiate it to get that slow.

07:53 So, y dash will be what we can apply here...

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