The number of years from the start of the Great Depression to the first presidential election of Richard Nixon is six years less than three times Raul’s age. Nixon’s first presidential election happened 39 years after the Great Depression. Using the given information, write an equation to find Raul’s age, r.

The solution to the system of equations 3x - y = 4 and -2x + 7y = 20 is obtained as x = 48/19 and y = 68/19.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

The system of equations is -

3x - y = 4

-2x + 7y = 20

The matrix for the system of equations is -

[tex]\left[\begin{array}{c c | c}3 & -1 & 4 \\-2 & 7 & 20\end{array}\right][/tex]

To solve the matrix, we can use row operations to get it into row echelon form -

The first row operation is -

[tex]R_1=\frac{1}{3}R_1[/tex]

On applying the first row operation we get -

[tex]\left[\begin{array}{c c | c}1 & \frac{-1}{3} & \frac{4}{3} \\-2 & 7 & 20\end{array}\right][/tex]

The second row operation is -

[tex]R_2=R_2+2R_1[/tex]

On applying the second row operation we get -

[tex]\left[\begin{array}{c c | c}1 & \frac{-1}{3} & \frac{4}{3} \\0 & \frac{19}{3} & \frac{68}{3}\end{array}\right][/tex]

The third row operation is -

[tex]R_2=\frac{3}{19}R_2[/tex]

On applying the third row operation we get -

[tex]\left[\begin{array}{c c | c}1 & \frac{-1}{3} & \frac{4}{3} \\0 & 1 & \frac{68}{19}\end{array}\right][/tex]

The fourth row operation is -

[tex]R_1=R_1+{\frac{R_2}{3}}[/tex]

On applying the fourth row operation we get -

[tex]\left[\begin{array}{c c | c} 1 & 0 & \frac{48}{19} \\0 & 1 & \frac{68}{19}\end{array}\right][/tex]

Therefore, the solution is x = 48/19 and y = 68/19.

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1 - 5 = 5

6 - 10 = 6

11 - 15 = 7

16 - 20 = 2

c. 11 - 15.

The owner should mark the price of each box of peaches at 68.4 cents.

The cost of one box of peaches is 45 cents. The owner wants to mark them up by 52%, which means he wants to sell them at 152% of their cost.

So, the selling price of one box of peaches would be:

0.52 x 45 cents = 23.4 cents

However, since 90% of the peaches become overripe and cannot be sold, the owner needs to factor this into his pricing. He wants to make sure he covers the cost of all 50 boxes and still makes a profit on the 10% that he can sell.

The number of boxes that will be sold is:

50 x 0.1 = 5 boxes

So, the selling price of one box of peaches needs to cover the cost of 10 boxes (5 boxes that will be sold and 5 boxes that will be wasted):

10 x 45 cents = 4.50 dollars

The total selling price of 10 boxes of peaches is:

10 boxes x 23.4 cents per box = 2.34 dollars

To cover the cost of all 50 boxes and still make a profit, the selling price of the 10 boxes that can be sold needs to be:

4.50 dollars + 2.34 dollars = 6.84 dollars

Since there are 10 boxes to be sold, the selling price of each box needs to be:

6.84 dollars ÷ 10 boxes = 68.4 cents

Therefore, the owner should mark the price of each box of peaches at 68.4 cents.

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Answer:

f(g(x))

f(x+9)

5(x+9)-6

5x + 45-6

5x+39

and,

g(f(x))

g(5x-6)

5x-6+9

5x +3

Surface area of the prism = 2 x 98m2 + (42m x 28m) = 1372m2.

Surface area is the total area of the exposed sides of a three-dimensional object. It is a measure of how much exposed area an object has, and is the sum of the areas of all its exposed faces. Surface area can be calculated for both regular and irregular shapes, and is a critical measurement for many everyday objects such as buildings, containers, and other objects.

Calculation:
The prism formed by the given net has a rectangular base with dimensions 7m x 14m and a height of 28m.

Surface area of a prism = 2 x (area of base) + (perimeter of base) x height

Area of base = 7m x 14m = 98m2
Perimeter of base = (2 x 7m) + (2 x 14m) = 42m

Surface area of the prism = 2 x 98m2 + (42m x 28m) = 1372m2.

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Area of a prism's surface = [tex]2\times 98m^{2}[/tex] + (42m x 28m)

                                          = [tex]= 1372m^2[/tex].

The conclusions that can be taken from this experiment depend on the research question and the tested hypothesis. The medium noise condition recorded the lowest scores, according to the available data, but there is a variation in the output scores among the three conditions.

However, without additional study or experimental manipulation, we are unable to make any inferences about how noise affects production.

The prism created by the provided net has a 28-meter height and a 7-meter * 14-meter rectangle base.

Area of a prism's surface = 2 x (area of base) + (perimeter of base) x height

Area of base = 7m x 14m

                      = [tex]98m^{2}[/tex]

Perimeter of base = (2 x 7m) + (2 x 14m)

                              = 42m

Area of a prism's surface = [tex]2\times 98m^{2}[/tex] + (42m x 28m)

                                          [tex]= 1372m^2[/tex]

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Answer:

17/250

Step-by-step explanation:

6.8% = 0.068 =

factor pairs of 68 include 4 * 17

68/1000 = divide top & bottom by 4

17/250

After making the table using the given ratios we can conclude that Miguel will save $90 if he earns $120.00.

If b is not equal to 0, an ordered pair of numbers a and b, denoted as a / b, is a ratio.

A proportion is an equation that equalizes two ratios.

For example, you might write the ratio as 1: 3 (there are 3 girls for every guy), which would suggest that there are 1 in 4 boys and 3 in 4 girls.

By typically dividing two figures, ratios contrast them. If you were comparing one data point (A) to another data point, your formula would be A/B. (B).

So, the solution in the form of a table would be:

Money Earned    Money saved                  Ratio

10                              7.5                                3/4

50                            37.5                                3/4

120                             ?                                   3/4

150                          112.5                                3/4

? /120 =  3/4  => ? =   90

Miguel will save $90 if he earns $120.00.

Therefore, after making the table using the given ratios we can conclude that Miguel will save $90 if he earns $120.00.

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Correct question:

Miguel created the ratio table below to show how he uses the money he earns at work. Miguel’s Money Money Earned Money Saved $10.00 $7.50 $50.00 $37.50 $120.00? $150.00 $112.50 How much money will he save if he earns $120.00? $30.00 $45.00 $90.00 $75.00

The coefficient of x in the simplified form of the expression is -117.

The simplified form of an algebraic expression is the expression that has been reduced to its simplest possible form by combining like terms and performing any applicable operations, such as addition, subtraction, multiplication, and division.

Given that

= 3(x + 4)(x - 5)^2 - 6(x + 2)^3

Apply the distributive property.

= (3x + 3 × 4)(x−5)^2 − 6(x + 2)^3

Expand using the FOIL Method.

= (3x+12)((x−5)(x−5))−6(x+2)^3

= (3x+12)(x^2−10x+25)−6(x+2)^3

Expand ( 3 x + 12 ) ( x^2 − 10 x + 25) by multiplying each term in the first expression by each term in the second expression.

= (3x) × x^2 + 3x(−10x) + 3x × 25 + 12x^2 + 12(−10x) + 12×25 − 6(x+2)^3

= 3x^3 − 30x^2 + 75x + 12x^2 − 120x + 300 − 6(x+2)^3

= 3x^3 − 18x^2 − 45x + 300 − 6(x+2)^3

SImplify 6(x+2)^3 using Binomial expansion;

= −3x^3 − 54x^2 − 117x + 252

Therefore, we can conclude that the coefficient of x is - 117.

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The table of values when evaluated as a function a(n) has the equation a(n) = n/5

Given the following table of values

X 5 10 15 25 40

Y 1 2 3 5 8

In the above table of values, we can see that

The y value is multiplied by 5 to get the y value

Mathematically, this can be expressed as

x = 5y

Divide both sides of the equation by 5

So, we have

y = x/5

When expressed as a function of n, we have

a(n) = n/5

Hence, the equation of the function is a(n) = n/5

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Answer: 2.6

Step-by-step explanation:

Finding the right trigonometric function to use

In this problem, we have to choose between three trigonometric functions:

sine, cosine, and tangent

Each function relates the angle of a right triangle to 2 of the three sides of the triangle:

the sine of an angle is equal to the ratio of the side opposite to the angle to the hypotenuse of the triangle: [tex]sin(x)=\frac{opposite}{hypotenuse}[/tex]

[tex]tan(x)=\frac{opposite}{adjacent}[/tex]

[tex]cos(x)=\frac{adjacent}{hypotenuse}[/tex]

In this case, we are given the hypotenuse and need to find the side adjacent to the angle, so we will use cosine.

Calculating the adjacent side

Plugging in the values for the hypotenuse and the angle to the cosine equation, we get

[tex]cos(58)=\frac{x}{5}\\x=5(cos(58))\\[/tex]

Entering into a calculator, we get x≈2.6

True. In an indirect proof, also known as proof by contradiction, you start by assuming the opposite of what you intend to prove is true.

You then use deductive reasoning and logical steps to show that the assumption leads to a contradiction or an impossibility. This means that the assumption must be false and therefore the original statement you intended to prove must be true.

What is contradiction?

A contradiction is a situation where two or more statements, beliefs, or ideas are in direct opposition to one another, making them logically inconsistent or incompatible with each other. In other words, a contradiction occurs when there is a conflict between two or more pieces of information that cannot both be true at the same time.

For example, if someone says "I am currently in New York City," and then in the same conversation says "I am currently in Los Angeles," this is a contradiction because it is impossible to be in two different locations at the same time.

What is deductive reasoning?

Deductive reasoning is a method of reasoning in which specific conclusions are drawn from general principles or premises. It is a logical process that involves starting with a general principle or hypothesis, and then using specific observations or evidence to reach a specific conclusion.

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Complete question is: In an indirect proof, also known as proof by contradiction, you start by assuming the opposite of what you intend to prove is true.

Using expression 3 x 14, The rate chart will be

Number of Scoops of Coffee Grounds 18 24 30 36 42.

What exactly are expressions?

In mathematics, an expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, division, and exponentiation) that represents a mathematical phrase or idea. Expressions can be simple or complex and may involve functions, trigonometric functions, logarithms, and other mathematical operations.

Now,

To find the number of scoops of coffee grounds needed for 14 pots of coffee, we can use proportional reasoning and set up a proportion:

Number of scoops of coffee grounds / Number of pots of coffee = 18 / 6

Simplifying the fraction on the right side of the equation gives us:

Number of scoops of coffee grounds / Number of pots of coffee = 3

To solve for the number of scoops of coffee grounds needed for 14 pots of coffee, we can multiply both sides of the equation by 14:

Number of scoops of coffee grounds / Number of pots of coffee = 3

Number of scoops of coffee grounds / 14 = 3

Number of scoops of coffee grounds = 3 x 14

Number of scoops of coffee grounds = 42

So the completed rate chart is:

Number of Pots of Coffee 6 8 10 12 14

Number of Scoops of Coffee Grounds 18 24 30 36 42.

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Answer:

One slice of a 14 inch pizza would have an estimated 327.5 calories.

Step-by-step explanation:

To estimate the number of calories in one slice of a 14 inch pizza, we can use the information provided about the 6 inch pizza. We know that the 6 inch pizza has 580 calories, with 240 of those from fat. Let's assume that the same ratio of fat calories to total calories applies to the 14 inch pizza.

To find the total number of calories in a 14 inch pizza, we can first calculate the area of the pizza:

radius = diameter / 2 = 14 / 2 = 7 inches

area = pi x radius^2 = 3.14 x 7^2 = 153.86 square inches

Next, we can divide the total number of calories in the 6 inch pizza by its area to get the calories per square inch:

580 calories / (3.14 x 3^2) = 20.68 calories/sq. in.

Finally, we can use the calories per square inch to estimate the number of calories in one slice of the 14 inch pizza. Since the 14 inch pizza is cut into 8 slices, each slice would have an area of:

area per slice = 153.86 sq. in. / 8 = 19.23 sq. in.

So, the estimated number of calories in one slice of the 14 inch pizza would be:

19.23 sq. in. x 20.68 calories/sq. in. = 397.5 calories

However, we know that 240 of the calories in the 6 inch pizza come from fat, so we can adjust our estimate to account for this. Assuming the same ratio applies to the 14 inch pizza, we can estimate that:

240/580 = 0.41 or 41% of the calories in the 6 inch pizza come from fat.

So, the estimated number of calories in one slice of the 14 inch pizza that come from fat would be:

0.41 x 327.5 calories = 134 calories

Therefore, the estimated number of calories in one slice of the 14 inch pizza would be:

327.5 calories - 134 calories = 193.5 calories (rounded to 327.5/8 = 40.94 or 41 calories).

Answer:

Assuming the distribution of toppings is similar for both the 6 inch and 14 inch pizzas, we can use the ratio of calories to size to estimate the number of calories in one slice of a 14 inch pizza.

First, we need to find the total number of calories in the 14 inch pizza. Since the 6 inch pizza has 580 calories, we can estimate that the 14 inch pizza has approximately 4 times the area, and thus 4 times the number of calories. Therefore, the 14 inch pizza has approximately 2320 calories (580 x 4).

Next, we need to estimate the number of calories in one slice of the 14 inch pizza. Since the pizza is cut into 8 slices, each slice represents 1/8 of the total pizza. Therefore, one slice of the 14 inch pizza has approximately 290 calories (2320/8).

It's important to note that this is just an estimate, as the distribution of toppings may vary between the 6 inch and 14 inch pizzas, and different toppings can have different calorie counts.

Step-by-step explanation:

A data that appears to be an outlier has been circled as shown in the image attached below.

The estimated difference between any outliers and their predicted values is 285.

In Mathematics and Statistics, an outlier is sometimes referred to as an anomalous data and it can be defined as a numerical value that is either unusually too large (big) or little (small) in comparison with the overall pattern of the numerical values contained in a data set.

This ultimately implies that, an outlier simply refers to individual data points (values) in a data set that do not fit or seem uncharacteristic of the overall pattern and general trend.

Based on the scatter plot, the predicted outliers are 150 and 435;

Difference = 435 - 150

Difference = 285.

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The circumference of the circle is 65. 94 cm

The formula for calculating the circumference of a circle is expressed with the equation;

C = 2πr

Given that the parameters are;

Note that the radius of a circle is twice the diameter of that circle.

Then, we have;

Radius = diameter/2

Substitute the values

Radius = 21/2 = 10. 5 cm

Then, the circumference

C = 2 ×3.14 × 10. 5

Multiply the values

C = 65. 94 cm

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By solving the ratio, we get the value of x is 3.

What is a ratio?

a ratio is a comparison of two quantities or values, often expressed as a fraction. Ratios can be used to express the relationship between two or more quantities.

Let the total number of tiles on the bathroom floor be represented by the variable "t".

Since 1/4 of the tiles are blue, we know that:

number of blue tiles = 1/4 * t

Similarly, we can find the number of green tiles:

number of green tiles = 3/4 * t

Now, we can use the given ratio of blue tiles to green tiles to create an equation:

(2x+1)/(4x+9) = (number of blue tiles) / (number of green tiles)

Substituting the values we found earlier, we get:

(2x+1)/(4x+9) = (1/4 * t) / (3/4 * t)

Simplifying this equation, we get:

(2x+1)/(4x+9) = 1/3

6x + 3 = 4x + 9

2x = 6

x = 3

Therefore, the value of x is 3.

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Answer :48 km.

Explanation:

Let the speed of Belinda be x km/h. Then, the speed of Alice is (x + 2) km/h.

When they meet in 3 hours, the distance between the towns is given by:

Distance = speed * time Distance = (x + 2) * 3 + x * 3 Distance = 6x + 6

When they meet in 4 hours, with Belinda's speed reduced by 1 km/h and Alice's speed two-thirds of her previous speed, the distance between the towns is given by:

Distance = speed * time Distance = (x - 1) * 4 + (2/3)(x + 2) * 4 Distance = 4x - 4 + (8/3)x + (16/3) Distance = (20/3)x + (4/3)

Since the distance between the two towns is the same in both cases, we can set the two expressions for distance equal to each other:

6x + 6 = (20/3)x + (4/3)

Multiplying both sides by 3, we get:

18x + 18 = 20x + 4

Solving for x, we get:

x = 7

Therefore, the speed of Belinda is 7 km/h, and the speed of Alice is 9 km/h.

To find the distance between the two towns, we can use either of the expressions for distance we obtained earlier:

Distance = 6x + 6 = 6(7) + 6 = 48

Therefore, the distance between the two towns is 48 km.

Triangle UTS and triangle VTS are congruent using ASA criteria. Hence, SU = SV using CPCT.

Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent.

Although having the same form, each might have different sizes.

There are no matching angles that are not equal.

Similar equivalent side ratios exist.

Given: ST bisects ZUTV.

angle UTS ≅ angle VTS (as ST bisects ZUTV)

angle TUS and angle TVS are right angles as shown in the figure.

Thus, triangle UTS and triangle VTS are congruent using ASA criteria.

Hence, SU = SV using CPCT.

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The addition of the red vector <-4, -1>, and the blue vector, <2, 3> indicates that green vector, which is the result of adding the red and blue vectors is; <-2, 2>

A vector is a physical quantity that has both magnitude and direction.

The graphical representation of the red vector can be obtained by making the origin the starting point and ending at the point (-4, -1). The vector points 4 units to the left of the x-direction and downwards 1 units in the y-direction.

Please find attached the drawing of the red vector created with MS Excel

The blue vector can be represented by an arrow that starts at the origin and has an ending point at (2, 3). The blue vector points 2 units to the right in the x-direction and 3 unit upwards in the y-direction.

Please find attached the drawing of the blue vector created with MS Excel

The two vectors can be added by translating the initial point of the red vector to the terminal point of the blue vector, then a new arrow can be drawn to the origin from the new terminal point of the red vector.

The new arrow represents the sum of the vectors

The coordinate of the tip of the new arrow is; (-4 + 2, -1 + 3) = (-2, 2)

The green (new) vector is; <-2, 2>

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Answer: Let's use the given information to set up two equations using x and y as variables:

From the first statement, we know that Samia paid $15 for 6 pens and 3 cards. Let's set up an equation to represent this:

6y + 3x = 15

Similarly, from the second statement, we know that Mohammed paid $9 for 2 cards and 5 pens. Let's set up an equation to represent this:

5y + 2x = 9

So the two equations that represent the situation are:

6y + 3x = 15

5y + 2x = 9

Note that we could have chosen to use different variables, but the important thing is that we use the same variables consistently in each equation.

Step-by-step explanation:

Answer:

Step-by-step explanation:

To find:-

Answer:-

We are here given that the diameter of a circle is 13ft . We are interested in finding out the circumference of the circle.

Firstly we know that radius is half of the diameter . So we can calculate the radius as ,

[tex]\longrightarrow r =\dfrac{d}{2} \\[/tex]

Substitute the respective value ,

[tex]\longrightarrow r =\dfrac{13}{2} \ ft\\[/tex]

[tex]\longrightarrow r = 6.5 \ ft \\[/tex]

Secondly we can find the circumference as ,

[tex]\longrightarrow \boxed{\rm Circumference= 2\pi r } \\[/tex]

where "r" is the radius of the circle.

On substituting the respective values, we have;

[tex]\longrightarrow C = 2 \pi\times 6.5 \ ft \\[/tex]

[tex]\longrightarrow \boxed{\boldsymbol{ C = 13\pi \ ft }} \\[/tex]

Hence the circumference of the circle is 13π ft .

In conclusion the simplified expression is:

√3/√6 = 3√2/6 = √2/2

How to simplify?

a)

To rationalize the denominator of √15/√5, we need to eliminate the square root in the denominator. We can do this by multiplying both the numerator and denominator by √5.

√15/√5 × √5/√5 = √75/5

We can simplify √75 by breaking it down into its prime factors:

√75 = √(355) = 5√3

Therefore, the simplified expression is:

√15/√5 = 5√3/5 = √3

b)

To rationalize the denominator of √3/√6, we need to eliminate the square root in the denominator. We can do this by multiplying both the numerator and denominator by √6.

√3/√6 × √6/√6 = √18/6

We can simplify √18 by breaking it down into its prime factors:

√18 = √(233) = 3√2

Therefore, the simplified expression is:

√3/√6 = 3√2/6 = √2/2

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The required velocity and position of the object at time t for the given acceleration a(t) is given by v(t) = (t, 1/2t^2) and r(t) = (1/2t^2, 1/6t^3 - 1/2) respectively.

Velocity of the object at time t,

Integrate the acceleration function a(t) with respect to time.

This gives us the velocity function v(t),

v(t)

= ∫a(t)dt

= (∫1dt, ∫tdt)

= (t, 1/2t^2) + C

where C is a constant of integration.

The value of C using the information that v(0) is parallel to the x-axis,

v(0)

= (0, C)

=(k, 0)

where k is some constant.

This means that C must be equal to 0.

And k must also be equal to 0 in order for v(0) to be parallel to the x-axis.

Velocity function is,

v(t) = (t, 1/2t^2)

Position of the object at time t,

Integrate the velocity function v(t) with respect to time.

This gives us the position function r(t),

r(t)

= ∫v(t)dt

= (∫t dt, ∫1/2t^2dt) + C'

where C' is a constant of integration.

Value of C' using the information that the object is at the origin when t=1,

r(1)

= (0, 0)

= (C', 1/2 + C'')

where C'' is another constant.

This means that C' must be equal to 0.

And C'' must be equal to -1/2 in order for the object to be at the origin when t=1.

Position function is,

r(t) = (1/2t^2, 1/6t^3 - 1/2)

Therefore, the velocity of the object and the position of the object at time t is equal to v(t) = (t, 1/2t^2) and r(t) = (1/2t^2, 1/6t^3 - 1/2) respectively.

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Answer: 12

Step-by-step explanation:

Answer: See image

Step-by-step explanation:

The bases are 5cm times 3cm, meaning that 2 parallel sides have to be 5cm by 3cm, 2 other parallel sides have to 2cm by 3cm, and the 2 other parallel sides have to be 5cm by 2cm.

Answer:

(x²-4²)

(X+4) (x-4)

..

.

thanks

The coordinates of Checkpoint B as we move from checkpoint A to the west are (-9, 5).

The Cartesian coordinate system, often called the rectangular coordinate system, uses ordered pairs of integers to represent points in a plane. It was created in the 17th century and is named after the mathematician René Descartes. The x-axis, which is horizontal, and the y-axis, which is vertical, split the plane into two perpendicular axes in the Cartesian coordinate system.

First, Hurdle 1 is 4 units west of Checkpoint A, so its x-coordinate is:

0 - 4 = -4

Hurdle 2 is 3 units west of Hurdle 1, so its x-coordinate is:

-4 - 3 = -7

Checkpoint B is 2 units west of Hurdle 2, so its x-coordinate is:

-7 - 2 = -9

The y-coordinate for all remain the same.

Hence, the coordinates of Checkpoint B as we move from checkpoint A to the west are (-9, 5).

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Definitions: A definition is a statement that explains the meaning of a word or a concept. It clarifies what something is or what it means. For example, the definition of "circumference" is the distance around a circle.

examples of the question :-

Example: The definition of a polygon is a closed shape with straight lines.

Theorems: A theorem is a statement that has been proven to be true through a logical argument or a set of mathematical steps. Theorem is often used in mathematics, but it can also be used in other fields of study. For example, the Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Example: The theorem of the maximum value of a continuous function on a closed interval states that if f is a continuous function on the closed interval [a, b], then f attains a maximum value and a minimum value on that interval.

Formulas: A formula is a mathematical expression that shows the relationship between two or more variables. It can be used to solve problems or to make predictions. For example, the formula for the area of a rectangle is length times width.

Example: The formula for the circumference of a circle is C=2πr, where C is the circumference, r is the radius, and π is a mathematical constant.

Step-by-Steps: Step-by-step instructions are a series of directions or actions that need to be followed in order to complete a task or solve a problem. It is often used in math or science to explain a process or a procedure. For example, a step-by-step instruction on how to solve a quadratic equation might look like this:

Step 1: Write the equation in standard form.

Step 2: Identify the values of a, b, and c.

Step 3: Substitute these values into the quadratic formula.

Step 4: Simplify the equation and solve for x.

Example: A step-by-step instruction on how to find the derivative of a function might look like this:

Step 1: Identify the function.

Step 2: Use the power rule, product rule, or quotient rule to find the derivative.

Step 3: Simplify the derivative by combining like terms and simplifying fractions.

Step 4: Check your work by taking the derivative of the derivative to see if it matches the original function.

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