Christa purchased ten bottles of shampoo in each of three different sizes: 312 ounces; 612 ounces; and 812 ounces. She used the expression (10×312)+(10×612)+(10×812) to show the total amount of shampoo purchased in ounces. Which expression is equivalent to the one Christa used?

The production figure to maximize the profit is is Z = 15x + 10y  subject to

2x + 3y ≤ 600 ; 2x + y ≤ 480  ; x,y≥ 0 .

In the question ,

it is given that ,

a company named Universal Electric manufactures sell two models of lamps that is Lamp X and Lamp Y  and

Whose profit is  $ 15 and $ 10 respectively.

let the number units the company manufactures of lamp X be = x units

let the number units the company manufactures of lamp Y be = y units

let the total profit be = Z ,

So , the profit to maximize be Z = 15x + 10y

worker A assembles Lamp X in 20/60 hours and Lamp Y in 20/60 hours ; and that he is available for 100 hours per month;

So , the inequality is

(20/60)x + (30/60)y ≤ 100.

20x + 30y ≤ 6000

2x + 3y ≤ 600

Similarly , worker B assembles Lamp X in 20/60 hours and Lamp Y in 10/60 hours ; and that he is available for 80 hours per month;

the inequality will be ,

(20/60)x + (10/60)y ≤ 80.

20x + 10y ≤ 4800

2x + y ≤ 480  .

Therefore , the LPP to maximize the profit is Z = 15x + 10y  subject to

2x + 3y ≤ 600 ; 2x + y ≤ 480  ; x,y≥ 0 .

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The possible values of B in the number line are option A, C, D and E

A number line is a visual representation of numbers on a straight line. This line is used to compare numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally or vertically. As we move towards the right side of a horizontal number line, the numbers increase; as we move towards the left, the numbers decrease.

In this problem, the value of B is between 3 and 2 since B is located between them.

Checking the given options;

A = 2.5

This is possible because B can be at the middle between 2 and 3

B = 2/5

This fraction isn't possible because the value of B is greater than this.

C = 5 / 2

This fraction is possible because the decimal value is 2.5 which is the same as option A

D = 25 / 10

This fraction is also possible because the decimal value is 2.5 which is same as option A and C

E = 2.49

This is also a likely possibility as this can be approximated to 2.5

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decision tree and neutral network

What is decision tree?

The most effective and well-liked technique for categorization and prediction is the decision tree. A decision tree is a type of tree structure that resembles a flowchart, where each internal node represents a test on an attribute, each branch a test result, and each leaf node (terminal node) a class label.

A model is an automatic flexible non linear regression model not interpretable and retains all predictor variables although it may limit the effect of some decision tree

A model is handles outliers well handle many potential predictor variable easy and short through predictor variables repeatedly to divide the data into group to best determine the response variable is  neutral network

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The value of f(2000 , 20) is $6640.24   and the interpretation is "if 2000 was invested initially then $6640.24 will be the amount after 20 years"  .

In the question ,

it is given that,

the future value (S) on investment earning is compounded continuously at 6% .

the function of the principal P and the length of time t that the principal has been invested is given by

S = f(P, t) = P×[tex]e^{0.06 \times t}[/tex]

to find f(2000 , 20) ,

we substitute P = 2000 and t = 20 ,

we get ,

f(2000 , 20) = 2000×[tex]e^{0.06 \times 20}[/tex]

Simplifying further ,

we get ,

= 6640.2338

≈ 6640.24

Its interpretation is "if 2000 was invested initially then $6640.24 will be the amount after 20 years .

Therefore , the value of f(2000 , 20) is $6640.24  .

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The function represents f ( x ) is f ( x ) = 32 ( 3x - 1 )

Given :

The table represents some points on the graph of linear function f.

Table :

x        -2         1        5       10

f ( x ) -224   64     448   928

take two points ( 5 , 448 ) and ( 1 , 64 )

slope m = 64 - ( 448 ) / 1 - ( 5 )

= 64 - 448 / 1 - 5

= -384 / -4

= 384 / 4

= 192 / 2

= 96

( 1 , 64 ) on

y -  64 = 96 ( x - 1 )

y - 64 = 96x - 96

y = 96x - 96 + 64

y = 96x - 32

take 32 as common

= 32 ( 3x - 1 )

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The semicircle above has a radius of r inches, The distance between the chord and the diameter in terms of r is √5 r/3 inches.

A chord of a circle can be defined as a line segment connecting any two points on the circumference. It should be noted that the diameter is the longest chord of a circle which passes through the center of the circle.

To find the distance required one can use either the Pythagorean theorem or the trigonometric ratios. Let the semicircle have center O. The diameter AB has length 2r. Because chord CD is 2/3 of the length of the diameter,

CD=2/3(2r) = 4r/3

let the distance between diameter AB and chord CD be x inches. To find x draw a right triangle connecting center O, the midpoint, E of chord CD, and point C . Using Pythagorean theorem here,

r² = x² + (CD/2)²

=> r² = x² + (2r/3)²

=> r² = x² + 4r²/9

=> x² = r² - 4r²/9 = 5r²/9

=> x = √5 r/3

Hence, the distance between the chord and the diameter in terms of r is √5r/3..

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The answer to this integration is - [tex]\frac{u^{7} }{28} + c[/tex] .

u = 3 + x⁴

differentiate both sides,

⇒ du = 0 + 4x³ dx

⇒ du = 4x³ dx

⇒[tex]dx = \frac{du}{4x^{3} }[/tex]   ..........(1)

The integration is :

[tex]\int x^{3} ( 3 +x^{4} )^{6} dx[/tex]

Substitute the values from equation (1)

= [tex]\int x^{3} u^{6} \frac{du}{4x^{3} }[/tex]

= [tex]\frac{1}{4} \int u^{6} du[/tex]

=[tex]\frac{1}{4} . \frac{u^{7} }{7} + c[/tex] ( c = integration constant )

= [tex]\frac{u^{7} }{28} + c[/tex]

Hence, the answer to this integration is - [tex]\frac{u^{7} }{28} + c[/tex] .

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11

According to question

-5xy²+7x²y+3y³-8x³

putting x=2 and y=3

⇒ -5(2)(3)²+7(2)²(3)+3(3)³-8(2)³

⇒-5(18)+7(12)+3(27)-8(8)

⇒-90+84+81-64

∴  11

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The correct question is find the value of equation if x=2 and y=3 , -5xy² + 7x²y+3y³ - 8x³

If x and y be  i.i.d. unif(0, 1), then the covariance of x + y and x - y is zero

Here  x and y be i.i.d. unif(0, 1)

The i.i.d stands for independent and identically distributed

unif means that the uniformly distributed

Here x and y are i.i.d unif (0,1)

The covariance is defined as the relationship between the two random variables . Therefore, variance of one variable is equal to the change in another variable

Here we have to find the covariance of x + y and x - y

According to the properties of covariance

Cov(x+ y, x - y) = Cov (x, x) + Cov(y, x) - Cov(x, y) - Cov(y, y) = 0

Because Cov(x, x) = Cov (y, y)

Cov(y, x) = Cov(x, y)

Therefore, the covariance of x+y and x - y is 0

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There is an imbalance between males and females, with a ratio of 1.5 men to every woman in the population.

Which age structure diagram represents a population's ages the best?

Hence, there is an imbalance between males and females, with a ratio of 1.5 men to every woman in the population.

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The tanθ  for the angle θ that lies in Quadrant III is √35

Trigonometry simply means the branch of mathematics that is concerned with functions of angles as well as their applications to calculation. It also deals with the relationship between ratios and their angles.

Given: The angle θ lies in Quadrant III

cosθ= −1/6 (adjacent/hypotenuse)

adjacent = -1 and hypotenuse = 6

We can draw the position of the angle as shown in the image attached.

Using Pythagoras' theorem:

opposite = √6²-(-1)² = √35

tanθ = (-√35) / (-1) = √35       (tanθ = opposite/adjacent)

Note: opposite in  Quadrant III is negative

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The closest projection will have a value of -29/360.

we will follow these steps to use the Gram-Schmidt orthonormalization procedure for producing an orthonormal basis for the subspace spanned by 1, x, and x²,

Set u₁ = 1, a unit vector.

Set v₂ = x and u₂ = v₂ / ||v₂||.

Set v₃ = x² - (x², u₁)u₁ - (x², u₂)u₂, and u₃ = v₃ / ||v₃||.

Thus we get the orthonormal basis for the subspace spanned by 1, x, and x² to be {u₁, u₂, u₃}.

Now we will use the formula for the projection of a vector onto a subspace to find the closest-point projection of x³ into this subspace,

proj_subspace(x³) = (x³, u₁)u₁ + (x³, u₂)u₂ + (x³, u₃)u₃

The inner products will be calculated as follows:

(x³, u₁) = ∫x³ X 1 dx = 1/4

(x³, u₂) = ∫ x³ X x dx = 1/5

(x³, u₃) = ∫ x³ * (x² - (x², u₁)u₁ - (x², u₂)u₂) dx

Here all the integrals will have limits of 0 and 1.

Hence we get

= 1/9 - 1/4 X 1/2 - 1/5 X 1/3

= 1/9 - 1/8 - 1/15

= (40 - 45 - 24)/360

= -29/360

Thus, the projection of x³ onto the subspace spanned by 1, x, and x² is:

proj_subspace(x³) = 1/4 X u₁ + 1/5 X u₂ + (1/9 - 1/4 X 1/2 - 1/5 X 1/3) X u₃

This projection is the closest point in the subspace to x³, in the sense that the distance between x³ and proj_subspace(x³) is minimized.

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Evaluation of the integral is  [tex]\frac{81}{16}[/tex]

In mathematics, an integral is either a definite value representing the region under a function's graph for a certain interval or a new function whose derivative represents the original function (indefinite integral). The fact that a definite integral of any function that can be integrated may be found using the indefinite integral and a corollary to the calculus fundamental theorem connects these two interpretations.

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The median represents here as, Option B; More  than half the police officers wrote this number of tickets.

Given,

A police department collected the statistics for the number of tickets police officers wrote each day over the last 6 weeks.

Mean =5.23

Median =3

Mode =2

We have to find the representation of median in this situation;

Median;-

The value that, when a dataset is arranged, falls exactly in the middle is the median. It is a measure of central tendency that distinguishes between the values' lowest and maximum 50%. Depending on whether you have an odd or an even number of data points, the processes for calculating the median change.

Here,

The median 3 represents that, More than half the police officers wrote this number of tickets.

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The interval containing the middle-most76% of sample means  [tex]102.836 < \mathrm{x} < 137.164[/tex].

As per the data given in the above question,

A distribution of values is normal with a mean of [tex]\mu=120[/tex].

A standard deviation of [tex]\sigma=21[/tex].

Drawing samples of size n=13.

The middle-most76% of sample means.

Find the interval containing the middle-most76% of sample means?

[tex]\mathrm{P}(-\mathrm{z} < \mathrm{x} < \mathrm{z})=0.76\\\mathrm{P}(\mathrm{x} < \mathrm{z})-\mathrm{P}(\mathrm{x} < -\mathrm{z})=0.76\\[/tex]

Further

[tex]\mathrm{P}(\mathrm{x} < -\mathrm{z})=\mathrm{P}(\mathrm{x} > \mathrm{z})[/tex]

[tex]\mathrm{P}(\mathrm{X} < \mathrm{Z})-\mathrm{P}(\mathrm{X} > \mathrm{Z})=0.76\\\mathrm{P}(\mathrm{X} < \mathrm{Z})-1+\mathrm{P}(\mathrm{X} < \mathrm{Z})=0.46\\2 \mathrm{P}(\mathrm{X} < \mathrm{Z})=1.76\\\mathrm{P}(\mathrm{X} < \mathrm{Z})=0.88[/tex]

[tex]= NORMSINV (0.73)\\\mathrm{Z}=0.613[/tex]

[tex]& \mathrm{X}=\mathrm{z} \sigma+\mu \\& \mathrm{X}=0.613 \times 28+120 \\& \mathrm{X}=137.164-\mathrm{x} & =\mu-\mathrm{z} \sigma \\-\mathrm{x} & =120-0.613 \times 28 \\-\mathrm{x} & =102.836[/tex]

Now the interval sample is as follow,

[tex]102.836 < \mathrm{x} < 137.164[/tex]

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capital accumulation is 3,973,910.14.

To evaluate the definite integral, we will have to find the indefinite integral. We use integral by substitution to find the indefinite integral.

Capital Accumulation = ∫ dI/dt dt

= ∫ 16,000t (t^2 + 2)^2dt

If u = t^2 + 2 then du = 2t dt so the integral is

Capital accumulation = ∫ 16,000t (t^2 + 2)^2 dt

= ∫ 16,000 × (u - 2) × u^2 du/2

= ∫ 16,000u^3 - 64,000u^2 + 64,000udu/2

= 8,000u^4/4 - 32,000u^3/3 + 32,000u^2/2 + C

= 2,000u^4 - 10,667.7u^3 + 16,000u^2 + C

Substituting u again gives us:

Capital Accumulation = 2,000(t^2 + 2)^4 - 10,667.7(t^2 + 2)^3 + 16,000(t^2 + 2)^2 + C

To evaluate the definite integral, we need to find the values ​​of this expression at t=0 and t=5 and subtract the two.

Capital Accumulation = [2,000(5^2 + 2)^4 - 10,667.7(5^2 + 2)^3 + 16,000(5^2 + 2)^2] - [2,000(0^2 + 2)^4 - 10,667.7(0^2 + 2)^3 + 16,000(0^2 + 2)^2]

= [2000(27)^4 - 10667.7(27)^3 + 16000(27)^2] - [2000(2)^4 - 10667.7(2)^3 + 16000(2)^2]

= 4,607,143.47 - 1,333,333.33 + 691,200

= 3,973,910.14

Therefore, the five-year capital accumulation is 3,973,910.14.

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They sold $479 in ticket sales and they had a total of 297 tickets sold, so they must have sold $479/297 = $<<479/297=1.61>>1.61 per ticket.

There are two types of tickets, adult tickets and student tickets, and adult tickets are $3, so the student tickets are $1.61 - $3 = $<<1.61-3=-1.39>>-1.39.

However, a ticket cannot be negative so we know that all of the tickets must have been adult tickets and they sold 297 tickets for $479, so they sold 297 tickets * $3/ticket = $<<297*3=891>>891 worth of adult tickets. Answer: \boxed{891}.

Answer:

Compounded Semiannually: [tex](1.03454)^t[/tex] and APR = 3.454%

Compounded Quarterly: [tex](1.03469)^t[/tex] and APR = 3.469%

Step-by-step explanation:

The Compound Interest Formula is as follows: [tex]A = P(1 + \frac{r}{n})^{nt}[/tex], which can also be written as: [tex]A = P((1 + \frac{r}{n})^n)^t[/tex], using properties of exponents. This is crucial since we want one value inside the parenthesis to calculate the APR (Annual Percentage Rate). Since we want the APR, the only thing we really care about is the [tex](1 + \frac{r}{n})^n[/tex], but this is going to be our APR + 1.

In the question, it's formatted as: [tex]b^t[/tex], and this inside part is going to be our b, so: [tex]b = (1 + \frac{r}{n})^n[/tex], and to calculate the APR we simply subtract 1.

Compounded semiannually means the interest is compounded twice a year, so n = 2. We are also given the interest to be 3.425% which in decimal form is 0.03425

Plugging this into the formula stated above we get: [tex]b = (1 + \frac{0.03425}{2})^2 \approx 1.03454[/tex]

So we have the form:

[tex](1.03454)^t[/tex]

With our APR simply being the base - 1:

[tex]APR \approx 0.03454[/tex]

We multiply this by 100 to get it in percentage form:

[tex]APR\approx 3.454\%[/tex]

Compounded quarterly means the interest is compounded four times a year, so n = 4. Plugging in known values we get:

[tex]b = (1 + \frac{0.03425}{4})^4 \approx 1.03469[/tex]

So we have the form:

[tex](1.03469)^t[/tex]

With our APR simply being the base - 1

[tex]APR \approx 0.03469[/tex]

Multiply this by 100 to get it in percentage form:

[tex]APR \approx 3.469\%[/tex]

Answer:

[tex]\begin{array}{|c|c|c|}\cline{1-3} \vphantom{\dfrac12} \sf koooAccount& b^t& \sf Annual\;Percentage\;rate \\\cline{1-3} \vphantom{\dfrac12} \sf compounded\;annually& (1.03425)^t& 3.425\%\\\cline{1-3} \vphantom{\dfrac12} \sf compounded\;semiannually & \left(1.03454\right)^t &3.454\% \\\cline{1-3} \vphantom{\dfrac12} \sf compounded\;quarterly & \left(1.03469\right)^t& 3.469\%\\\cline{1-3} \vphantom{\dfrac12} \sf compounded\; daily& \left(1.03484\right)^t& 3.484\%\\\cline{1-3} \end{array}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

From inspection of the compound interest formula:

[tex]\implies b^t=\left[\left(1+\dfrac{r}{n}\right)^n\right]^t[/tex]

To find the annual percentage rate:

[tex]\implies APR=(b-1)\times 100[/tex]

Given:

Substitute the values into the formula for [tex]b^t[/tex]:

[tex]\implies b^t=\left[\left(1+\dfrac{0.03425}{1}\right)^1\right]^t[/tex]

[tex]\implies b^t=\left(1.03425\right)^t[/tex]

Therefore, the annual percentage rate is:

[tex]\implies APR=(1-1.03425) \times 100[/tex]

[tex]\implies APR=0.03425 \times 100[/tex]

[tex]\implies APR=3.425\%[/tex]

Given:

Substitute the values into the formula for [tex]b^t[/tex]:

[tex]\implies b^t=\left[\left(1+\dfrac{0.03425}{2}\right)^2\right]^t[/tex]

[tex]\implies b^t=\left(1.03454326...\right)^t[/tex]

[tex]\implies b^t=\left(1.03454\right)^t[/tex]

Therefore, the annual percentage rate is:

[tex]\implies APR=(1-1.03454) \times 100[/tex]

[tex]\implies APR=0.03454 \times 100[/tex]

[tex]\implies APR=3.454\%[/tex]

Given:

Substitute the values into the formula for [tex]b^t[/tex]:

[tex]\implies b^t=\left[\left(1+\dfrac{0.03425}{4}\right)^4\right]^t[/tex]

[tex]\implies b^t=\left(1.03469241...\right)^t[/tex]

[tex]\implies b^t=\left(1.03469\right)^t[/tex]

Therefore, the annual percentage rate is:

[tex]\implies APR=(1-1.03469) \times 100[/tex]

[tex]\implies APR=0.03469 \times 100[/tex]

[tex]\implies APR=3.469\%[/tex]

Given:

Substitute the values into the formula for [tex]b^t[/tex]:

[tex]\implies b^t=\left[\left(1+\dfrac{0.03425}{365}\right)^{365}\right]^t[/tex]

[tex]\implies b^t=\left(1.03484162...\right)^t[/tex]

[tex]\implies b^t=\left(1.03484\right)^t[/tex]

Therefore, the annual percentage rate is:

[tex]\implies APR=(1-1.03484) \times 100[/tex]

[tex]\implies APR=0.03484 \times 100[/tex]

[tex]\implies APR=3.484\%[/tex]

Note: There is an error in the given table:  the given "compounded daily" rate is incorrect.

The monthly payment required into the annuity with an APR of 8.0% compounded annually and you wish to reach your savings goal in 30 years is $156.19.

The monthly payment is required for 360 months (30 years x 12).

We can compute this periodic payment using an online finance calculator by inputting the parameters below:

N (# of periods) = 360 months (30 years x 12)

I/Y (Interest per year) = 8%

PV (Present Value) = $0

FV (Future Value) = $220,000

P/Y (# of periods per year) = 12

C/Y (# of times interest compound per year) = 1

Results:

Monthly Payments (PMT) = $156.19

Sum of all periodic payments = $56,228.40

Total Interest = $163,771.60

Thus, to reach a future value of $220,000, the monthly annuity deposit should be $156.19 for 30 years at 8%.

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Assuming no employees are subject to ceilings for their earnings IS $7,450

The computation of Salaries Payable is shown below:-

Gross Payroll $10,000

Less:

Federal income tax withheld $1,800

Social security rate at 6% -$600

Medicare rate at 1.5% -$150

Total deduction -$2,550

Salary Payable $7,450

We simply deduct three above taxes from the gross payroll so that we get to know for the salary payable amount

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The value of each company's product sold price is given as follows:

The values of each company's product sold price is obtained using a system of equations, for which the variables are defined as follows:

The sold sum of both the two companies salaries are $44,000,000, hence:

x + y = 44000000.

The sold price of company x is 1000 more than the other, hence:

x = 1000 + y.

Then the sold price of company y can be found replacing the second equation into the first, as follows:

1000 + y + y = 44000000

2y = 44000000 - 1000

y = (44000000 - 1000)/2

y = $21,999,500.

The sold price for company x is then calculated as follows:

x = 1000 + y = 1000 + 21999500 = $22,000,500.

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The critical values for the following sample are ±2.0064.

Here it is given that α = 0.05,

To calculate the critical values we will use the t-statistic.

Here n₁ = 25 and n₂ = 30

Hence, the degrees of freedom

d.f = n₁ + n₂ - 2           [Since it's a 2-sample test]

= 25 + 30 - 2

= 53

Now our significance level is  α = 0.05,

To know the critical value we will make use of a t-distribution table and look up the value corresponding to a 0.05 significance level and 53 degrees of freedom.

Here we will get the critical values ±2.0064

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Long division can be used to determine the  absolute value of x[0], x[-1], and x[-2] when the roc is known. To do this, divide the roc (113) by the coefficient of the highest power in the polynomial (x2). Then, divide the remainder (3) by the coefficient of the next highest power (x), which gives x[-1] = 3. Finally, divide the remainder (8) by the coefficient of the lowest power (1), which gives x[-2] = 8.

To use long division to determine the values of x[0], x[-1], and x[-2], begin by dividing the roc (113) by the coefficient of the highest power in the polynomial (x2). This gives a result of 113/x2 = 10. This means that x[0] = 10. This number is then used as the divisor for the next step, which is to divide the remainder (3) by the coefficient of the next highest power (x). This gives a result of 3/x = 3, so x[-1] = 3. Finally, divide the remainder (8) by the coefficient of the lowest power (1), which gives 8/1 = 8, so x[-2] = 8. This process can be repeated to determine the values of x[n] for any polynomial.

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Answer:

60% because the total should be 100% and if you take 40% away you should be left with 60%

Answer:

x=9

Step-by-step explanation:

x=9

To find out what is a solution for x<8  you have to plug in all the answer choices for x.

When you plug in -3, you get -3<8 and 8 is bigger than -3, so then that makes it a solution because -3 is less than 8 and when you plug it in it sets the equation right.

When you plug in 7.95, you get 7.95<8 and 8 is bigger than 7.95 by a few but it still is bigger so then it's a solution because 7.95 is less than 8 making it a solution.

When you plug in 4.3, you get 4.3<8 and 8 is bigger than 4.3 so then it's  a solution because it sets the equation right.

When you plug in 9, you get 9<8 and that's not right because 8 isn't bigger than 9, setting this answer choice as a non-solution to the inequality.

(Hope you understand this..)

Answer:

Step-by-step explanation:

probability ≤|1|

or -1≤ probability ≤1

Answer:

No

Step-by-step explanation:

The sum of the probabilities in a probability distribution is always 1.

Answer:

Step-by-step explanation:

ok

alr

Answer:

below

Step-by-step explanation:

Largest angle = largest opposite side  ,  smallest angle = smallest opposite side

EF < ED <DF

If the perimeter of the rectangle is 48 cm, its length and width is equal to 16 cm and 8 cm respectively.

Mathematically, the perimeter of a rectangle can be calculated by using this equation;

P = 2(L + W)

Where:

Since the length is twice the​ width, we have;

L = 2W

Substituting the given parameters into the perimeter formula, we have;

48 = 2(2W + W)

48 = 6W

W = 48/6

Width, W = 8 cm.

For the length of this rectangle, we have;

L = 2W

L = 2(8)

L = 16 cm.

Read more on perimeter of a rectangle here: brainly.com/question/17360327

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Answer:

Step-by-step explanation:

You want to rewrite f(x) and differentiate it for f(x) = 6·ln(8/x).

The log of a ratio is the difference of logs:

  ln(a/b) = ln(a) -ln(b)

Using this to rewrite f(x), we have ...

  f(x) = 6ln(8/x) = 6(ln(8) -ln(x))

  f(x) = 6·ln(8) -6·ln(x)

The derivative of the constant is zero, so ...

  f'(x) = -6/x