Given cos θ=-15/17 and 180°<θ<270° , find the exact value of each expression. cosθ/2

* When x = 16 and dx/dt = 4, then dy/dt = 1/(2√16) = 1/8.

* When x = 25 and dy/dt = 3, then dx/dt = 2√25 * 3 = 15.

* The first equation is found by differentiating y = √x with respect to t.

* The second equation is found by using the chain rule.

Here are the steps to find dy/dt:

1. Start by differentiating y = √x with respect to x.

2. The derivative of √x is 1/(2√x).

3. Multiply the derivative by dx/dt to get dy/dt.

Here are the steps to find dx/dt:

1. Start by differentiating y = √x with respect to t.

2. The derivative of √x is 1/(2√x).

3. Multiply the derivative by dy/dt to get dx/dt.

Therefore:  When x = 16 and dx/dt = 4, then dy/dt = 1/(2√16) = 1/8.

                   When x = 25 and dy/dt = 3, then dx/dt = 2√25 * 3 = 15.

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The ordered pair (3, 2) satisfies inequality B and C but not inequality A.

To determine which inequalities A, B, and C the ordered pair (3, 2) satisfies, we can substitute the values of x and y into each inequality and check if the statement holds true.

For inequality A: x + y ≤ 2

Substituting x = 3 and y = 2:

3 + 2 ≤ 2

5 ≤ 2

Since 5 is not less than or equal to 2, the ordered pair (3, 2) does not satisfy inequality A.

For inequality B: y ≤ (3/2)x - 1

Substituting x = 3 and y = 2:

2 ≤ (3/2)(3) - 1

2 ≤ (9/2) - 1

2 ≤ 4.5 - 1

2 ≤ 3.5

Since 2 is less than or equal to 3.5, the ordered pair (3, 2) satisfies inequality B.

For inequality C: y > -(1/3)x - 2

Substituting x = 3 and y = 2:

2 > -(1/3)(3) - 2

2 > -1 - 2

2 > -3

Since 2 is greater than -3, the ordered pair (3, 2) satisfies inequality C.

Therefore, the ordered pair (3, 2) satisfies inequality B and C but not inequality A.

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The distance between the foci of the ellipse is 12 units.

To find the distance between the foci of an ellipse, we can use the relationship between the lengths of the major and minor axes. The formula is given as:

c = √(a^2 - b^2)

Here, "c" represents the distance between the center of the ellipse and each focus, "a" represents half the length of the major axis, and "b" represents half the length of the minor axis.

In this case, the lengths of the major and minor axes are given as 30 and 18 respectively. So, a = 30/2 = 15 and b = 18/2 = 9.

Plugging these values into the formula, we have:

c = √(15^2 - 9^2)

c = √(225 - 81)

c = √144

c = 12

Therefore, the distance between the foci of the ellipse is 12 units.

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The formula based on the Pythagorean theorem is:

c² = a² + b²

To find the area of a right triangle using the Pythagorean theorem, you need the lengths of two sides of the triangle, one of which must be the base or height.

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.

Let's assume that the lengths of the two sides of the right triangle are a and b, and the hypotenuse is c. The formula based on the Pythagorean theorem is:

c² = a² + b²

To find the area, you need the base (b) and height (a) of the triangle. Since the base and height are the two legs of the right triangle, you can rearrange the Pythagorean theorem formula to solve for one of them:

a = √(c² - b²) or b = √(c² - a²)

Once you have the base and height, you can calculate the area using the formula:

Area = (1/2) * base * height

Substitute the values of the base and height into the formula to find the area of the right triangle.

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Answer:

1 of 4

Step-by-step explanation:

Answer:

15

Step-by-step explanation:

_×_=+

.................

The answer is:

15

Work/explanation:

Subtracting a negative is the same as adding a positive:

[tex]\sf{a-(-b)=a+b}[/tex]

Similarly,

[tex]\sf{5-(-10)=5+10=15}[/tex]

Hence, the answer is 15.

Sequences are functions defined on a subset of the integers, often with a recursive definition.

Sequences are mathematical objects that represent ordered lists of numbers. They can be thought of as functions whose domain is a subset of the integers. A sequence is typically defined recursively, where each term is determined by previous terms in the sequence. This recursive definition allows us to generate the terms of the sequence by applying a specific rule or formula.

Sequences are widely used in mathematics and various fields of science. They have applications in areas such as number theory, calculus, statistics, and computer science. Understanding the properties and behavior of sequences is essential in analyzing patterns, making predictions, and solving problems.

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Answer:

60 seconds          180 seconds

------------------- = ------------------

1 minute                x minutes

Step-by-step explanation:

To convert seconds to minutes, we need to use the conversion factor

60 seconds = 1 minute

60 seconds          180 seconds

------------------- = ------------------

1 minute                x minutes

The median number of adoptions is higher than the mean number of adoptions.

The histogram shows that the number of adoptions is skewed to the right. This means that there are a few regions with a very high number of adoptions, which is pulling the mean up. However, the median is not affected by outliers, so it remains closer to the center of the distribution.

In this case, the median number of adoptions is 58.5, while the mean number of adoptions is 51.27. This means that half of the regions had more than 58.5 adoptions, while half had less.

Here is a table showing the mean and median number of adoptions for each region:

| Region | Number of adoptions | Mean | Median |

|---|---|---|---|

| 1 | 100 | 51.27 | 58.5 |

| 2 | 95 | 51.27 | 58.5 |

| 3 | 90 | 51.27 | 58.5 |

| ... | ... | ... | ... |

| 45 | 55 | 51.27 | 58.5 |

| 46 | 55 | 51.27 | 58.5 |

| 47 | 55 | 51.27 | 58.5 |

| 48 | 55 | 51.27 | 58.5 |

As you can see, the mean and median are the same for all 48 regions. However, the mean is pulled up by the few regions with a very high number of adoptions. The median, on the other hand, is not affected by outliers, so it remains closer to the center of the distribution.

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Answer:

---------------------

In the given diagram, the central angle has same measure as given arc.

Hence we can set up the following equation:

The matching choice is D.

Answer is.

Step-by-step explanation:

Here we are given central angle of the circle is 85°

Length of arc is 4x - 5.

Since measure of central angle is equal to measure of an arc of the circle.

Then,

4x + 5 = 85

4x = 85 - 5

4x = 80

x = 80/4

x = 20

So, the value of x is 20

The explicit bijection is f(x) = (x - a) / (b - a)

To find an explicit bijection between the intervals [a, b) and (0, 1), use a linear transformation and scaling. Let's denote the bijection as f: [a, b) -> (0, 1).

First, shift the interval [a, b) to start from 0. We can achieve this by subtracting 'a' from each element in the interval. So, the shifted interval becomes [0, b - a).

Next, scale the interval [0, b - a) to (0, 1). To do this, we divide each element by the length of the interval (b - a). So, the scaled interval becomes (0, 1/(b - a)).

Finally, define the bijection f as follows:

f(x) = (x - a) / (b - a)

Let's verify that f is a bijection:

1. Injective (One-to-One):

Suppose f(x₁) = f(x₂) for some x₁, x₂ ∈ [a, b). Then, we have:

(x₁ - a) / (b - a) = (x₂ - a) / (b - a)

Cross-multiplying, we get:

(x₁ - a)(b - a) = (x₂ - a)(b - a)

Expanding and simplifying:

x₁(b - a) - a(b - a) = x₂(b - a) - a(b - a)

x₁(b - a) = x₂(b - a)

x₁ = x₂

Therefore, f is injective.

2. Surjective (Onto):

Let y ∈ (0, 1). We need to show that there exists an x ∈ [a, b) such that f(x) = y. Solving for x, we have:

(x - a) / (b - a) = y

Cross-multiplying, we get:

x - a = y(b - a)

Rearranging, we have:

x = y(b - a) + a

Since y ∈ (0, 1), we have 0 < y(b - a) < b - a. Therefore, x ∈ [a, b).

Thus, for any y ∈ (0, 1), find an x ∈ [a, b) such that f(x) = y.

Hence, f is surjective.

Since f is both injective and surjective, it is a bijection between [a, b) and (0, 1).

Therefore, the explicit bijection is:

f(x) = (x - a) / (b - a)

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The expression  [tex]sin 2\theta cos\theta + cos 2\theta sin \theta[/tex]  can be rewritten as [tex]sin \theta (2cos^2 \theta - sin \theta)[/tex] as a trigonometric function of a single-angle measure.

To rewrite the expression [tex]sin 2\theta cos\theta + cos 2\theta sin \theta[/tex] as a trigonometric function of a single angle measure, we can use trigonometric identities to simplify it.

Using the double angle formula for sine ([tex]sin 2\theta = 2sin \theta cos \theta[/tex]) and the double angle formula for cosine ([tex]cos 2\theta = cos^2\theta - sin^2 \theta[/tex]), we can rewrite the expression as:

[tex]2sin \theta cos^2 \theta - sin^2 \theta cos \theta + sin \theta cos^2 \theta[/tex]

Now, we can factor out [tex]sin \theta[/tex] and [tex]cos \theta[/tex]:

[tex]sin \theta (2cos^2 \theta - sin \theta) + cos \theta (sin \theta cos \theta)[/tex]

Simplifying further:

[tex]sin \theta (2cos^2 \theta - sin \theta) + cos \theta (sin \theta cos \theta)\\= sin \theta (2cos^2 \theta - sin \theta) + cos^2 \theta sin \theta[/tex]

Now, we can rewrite the expression as a trigonometric function of a single angle measure:

[tex]sin \theta (2cos^2 \theta - sin \theta) + cos^2 \theta sin \theta\\= sin \theta (2cos^2 \theta - sin \theta + cos^2 \theta)\\= sin \theta (cos^2 \theta + cos^2 \theta - sin \theta)\\= sin \theta (2cos^2 \theta - sin \theta)[/tex]

Therefore, the expression  [tex]sin 2\theta cos\theta + cos 2\theta sin \theta[/tex]  can be rewritten as [tex]sin \theta (2cos^2 \theta - sin \theta)[/tex] as a trigonometric function of a single angle measure.

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If Plane Z is an angle bisector of ∠ K J H, KJ ≅ HJ then MH ≅ MK.

Given that Plane Z is an angle bisector of ∠KJH.

By definition, since Plane Z is an angle bisector, it divides ∠KJH into two congruent angles: ∠MJH and ∠MKH.

Given that KJ ≅ HJ.

Using the ASA congruence criterion, we can conclude that ∆MJH is congruent to ∆MKH because they share an angle (∠MJH ≅ ∠MKH), the side MJ is common, and KJ ≅ HJ.

By the corresponding parts of congruent triangles, we can deduce that the corresponding sides MH and MK are congruent in the congruent triangles ∆MJH and ∆MKH, resulting in MH ≅ MK.

Statements                                                   Reasons

1. Plane Z is an angle bisector of ∠KJH   Given

2. ∠MJH ≅ ∠MKH                                           Definition of angle bisector

3. KJ ≅ HJ                                                    Given

4. ∆MJH ≅ ∆MKH                                 Angle-Side-Angle (ASA) congruence

5. MH ≅ MK                        Corresponding parts of congruent   triangles

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Optimal allocation of goods x1 and x2, given the utility function U(x1, x2) = x1^ρ + x2^ρ, as functions of prices p1 and p2 and income m, is given by x1* = [(p1/m)^(1/ρ)] * (U/m) and x2* = [(p2/m)^(1/ρ)] * (U/m).

These formulas allow for the determination of the optimal quantities based on the prices and income level. In this case, the optimal allocation of goods x1 and x2 depends on the relative prices of the goods (p1 and p2) and the level of income (m). The exponents ρ determine the level of substitutability or complementarity between the goods. When the prices and income are given, the formulas for x1* and x2* can be used to calculate the optimal quantities.

These formulas allow for the determination of the optimal quantities based on the prices and income level.

By taking the partial derivatives of the utility function with respect to x1 and x2 and setting them equal to zero, we find the values that maximize the utility given the constraints of prices and income. The exponents ρ in the utility function represent the degree of preference for each good, determining whether they are substitutes or complements.

The formulas for x1* and x2* indicate that the optimal quantities are determined by the ratios of the prices and income raised to the power of 1/ρ. These ratios reflect the relative affordability of the goods and their importance in the overall utility calculation. By plugging in the given prices and income, one can calculate the optimal values of x1 and x2, providing a solution for maximizing utility under the given conditions.

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The reference number for t is 13π/4, and the terminal point determined by t is (r, s), where r and s are the x-coordinate and y-coordinate of the terminal point, respectively.

In the polar coordinate system, a reference number represents an angle measured counterclockwise from the positive x-axis to the terminal side of the angle. The reference number provides information about the position of the terminal point on the unit circle.

To find the terminal point determined by t = 13π/4, we can use the unit circle and the reference number. The reference number 13π/4 indicates that the terminal side of the angle intersects the unit circle at a point that is 13π/4 radians or 292.5 degrees counterclockwise from the positive x-axis.

Since the unit circle has a circumference of 2π, an angle of 13π/4 is equivalent to an angle of 5π/4, which is 45 degrees more than a full revolution. Therefore, the terminal point determined by t = 13π/4 is located at an angle of 45 degrees (or π/4 radians) counterclockwise from the positive x-axis on the unit circle.

In summary, the reference number for t is 13π/4, and the terminal point determined by t = 13π/4 is located at an angle of 45 degrees (or π/4 radians) counterclockwise from the positive x-axis on the unit circle.

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The solution of equation y/5 + 4 = 9 is,

y = 25

We have to give that,

An expression to simplify,

y/5 + 4 = 9

Now, Combine like terms of the expression and find the value of y as,

y/5 + 4 = 9

Subtract 4 both side,

y/5 + 4 - 4 = 9 - 4

y/5 = 5

Multiply by 5 both side,

y/5 x 5 = 5 x 5

y = 25

Therefore, The solution is, y = 25

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The first difference of this function is,

d = 1

We have to give that,

The outputs for a certain function are 1,2,4,8,16,32, and so on.

Here, the outputs are,

⇒ 1, 2, 4, 8, 16, 32

Hence, the first difference of this function is,

d = 2 - 1

d = 1

Therefore, The solution for the first difference of this function is,

d = 1

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The surface area and volume of a hockey puck include the following:

Volume = 7.065 in³.

Surface area = 23.6 in².

In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

Radius = diameter/2 = 3/2 = 1.5 inches.

By substituting the given side lengths into the volume of a cylinder formula, we have the following;

Volume of hockey puck = 3.14 × (3/2)² × 1

Volume of hockey puck = 7.065 in³.

By substituting the given parameters into the formula for the surface area (SA) of a cylinder, we have the following;

Surface area = 2πrh + 2πr²

Surface area = 2π(3/2)(1) + 2π(3/2)²

Surface area = 23.6 in².

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When x and y vary inversely, their product remains constant. If x is 30 when y is 2, then when x is 5, y will be 12.

If x and y vary inversely, it means that their product remains constant. Mathematically, this can be expressed as:

x * y = k,

where k is the constant of variation.

That x = 30 when y = 2, we can substitute these values into the equation to find the value of k:

30 * 2 = k,

k = 60.

Now we can use this value of k to find y when x = 5:

5 * y = 60,

y = 60 / 5,

y = 12.

Therefore, when x = 5, y will be equal to 12.

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Jen can make sure that the canvas will be a square by measuring the lengths of the four sides of the frame. If all four sides have equal lengths, then the frame is a square.

Square is a polygon with all four sides equal and all angles measuring 90 degree. So if any polygon satisfies this criteria it is a square.

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The total cost for both the plans are same and the amount of money is $2600.

To determine if the total cost for renting furniture for x months is the same for both plans and the amount of money, we need to set up equations for each plan and solve for x.

For Plan A:

Total cost = $1100 + $125 per month

The equation for Plan A's total cost is: [tex]C_A = 1100 + 125x[/tex]

For Plan B:

Total cost = $200 + $200 per month

The equation for Plan B's total cost is: [tex]C_B = 200 + 200x[/tex]

To find the value of x when the total costs are the same, we can set up an equation and solve for x:

1100 + 125x = 200 + 200x

Subtract 125x and 200x from both sides:

1100 - 200 = 200x - 125x

900 = 75x

Divide both sides by 75:

900 / 75 = x

12 = x

So, when x = 12 months, the total costs for both plans are the same.

To find the amount of money, we can substitute x = 12 into either equation. Let's use Plan A:

[tex]C_A = 1100 + 125x\\C_A = 1100 + 125(12)\\C_A = 1100 + 1500\\C_A = 2600\\[/tex]

Therefore, the amount of money is $2600.

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The water level in the reservoir is currently at -8 feet. It dropped from an initial level of -1 foot to the current level over the course of the next 3 months.

The water level in the reservoir is currently -8 feet. Over the next 3 months, it drops 8 times as low as it was to start.

To find the current water level, we need to calculate how much it dropped by multiplying the initial level by 8.

Let's start by finding the initial water level. Since the initial level is 8 times higher than the current level, we can divide the current level by 8 to find the initial level.

-8 feet / 8 = -1 foot

So, the initial water level was -1 foot.

Now, we need to calculate the current water level. We know that it dropped 8 times as low as the initial level. To find the current level, we multiply the initial level by 8.

-1 foot * 8 = -8 feet

Therefore, the current water level is -8 feet.

In summary, the water level in the reservoir is currently at -8 feet. It dropped from an initial level of -1 foot to the current level over the course of the next 3 months.

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Based on the given information, you can borrow approximately $89,227.46 for the purchase of your first home if you can afford to make monthly payments of $500 with a 6% annual interest rate on a 30-year mortgage.

APR = [tex](FV / PV)^1^/^n[/tex] - 1

Where:

APR = Annual Percentage Rate

FV = Future Value ($2,500,000)

PV = Present Value or initial investment ($800,000)

n = number of years (10 years)

Plugging in the values, we have: APR = [tex]($2,500,000 / $800,000)^1^/^1^0[/tex] - 1

Calculating this expression, we find: APR ≈ 0.1153 or 11.53%

Therefore, the annual percentage rate of return on your investment in the cattle ranch is approximately 11.53%.

Regarding your question about how much you can borrow for the purchase of your first home, given that you can afford to make monthly payments of $500 and the annual interest rate is 6% on a 30-year mortgage, we can use the formula for loan amount calculation:

Loan Amount = (Monthly Payment / Monthly Interest Rate) * (1 - [tex](1 + Monthly Interest Rate)^-^N^u^m^b^e^r^ o^f^ M^o^n^t^h^s[/tex])

Where:

Monthly Payment = $500

Annual Interest Rate = 6% or 0.06

Monthly Interest Rate = Annual Interest Rate / 12

Number of Months = 30 years * 12 months

Plugging in the values, we have:

Loan Amount = ($500 / (0.06 / 12)) * (1 - (1 + [tex](0.06 / 12))^-^3^0^ *^ 1^2[/tex])

Calculating this expression, we find: Loan Amount ≈ $89,227.46

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Answer:

12,468.94[tex]yd^{3}[/tex]

Step-by-step explanation:

The volume of a cylinder is the area of the base times the height

a = (AB xH)

a = [tex]\pi r^{2}[/tex]x h

a = (3.14)[tex](19^{2})[/tex](11)

a = (3.14)(361)(11)

a = 12468.94

Helping in the name of Jesus.

The value of sec 195° rounded to the nearest thousandth is approximately -1.084.

The secant function (sec) is the reciprocal of the cosine function (cos). To find the value of sec 195°, we need to calculate the value of cos 195° and then take its reciprocal. Using a calculator, we find that cos 195° is approximately -0.087.

Now, to find sec 195°, we take the reciprocal of -0.087, which gives us approximately -1.084. Therefore, the value of sec 195° rounded to the nearest thousandth is approximately -1.084.

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(a) Alice's strategy is not a winning strategy. Bob can always win against this strategy by mirroring Alice's moves. Whenever Alice takes a square from the top row, Bob takes the corresponding square from the bottom row, and vice versa. By doing so, Bob can always ensure that he takes the last remaining square, thereby winning the game.

In the game of CHOMP, the player who takes the last remaining square loses. Alice's strategy of always taking a single square from the top row if possible, or from the bottom row if the top row is all gone, does not guarantee a win. Bob can exploit this strategy by mirroring Alice's moves. This means that whenever Alice takes a square from the top row, Bob takes the corresponding square from the bottom row, and vice versa.

By employing this strategy, Bob can ensure that he takes the last remaining square. For example, if Alice takes the square in the top-left corner, Bob will take the corresponding square in the bottom-left corner. This leaves a smaller board of 1×4. Bob can continue mirroring Alice's moves until there is only one square left, which he will take, winning the game. Therefore, Bob has a winning strategy against Alice's strategy in CHOMP.

(b) If Bob goes first and Alice still uses the same strategy, Bob will always win the game. Bob can adopt the strategy of always taking a single square from the top row if possible, or from the bottom row if the top row is all gone, just like Alice's strategy. Since Bob moves first, he can make the same move as Alice would have made in the first turn. By doing so, Bob puts himself in the same advantageous position that Alice would have been in. From this point on, Bob can employ the mirroring strategy mentioned above and guarantee a win.

(c) If both players use the same strategy of always taking a single square from the top row if possible, or from the bottom row if the top row is all gone, the game will end in a draw. Both players will continue mirroring each other's moves, resulting in a symmetrical game progression. Eventually, all squares will be taken, and no player will be left with the last square. Hence, there is no winning strategy for either player when both follow this strategy.

Bonus: A winning strategy for the first player can be achieved by modifying the initial strategy. The first player should intentionally leave a specific square for the opponent to take, such that it leads to a losing position. By strategically choosing their moves, the first player can force the second player into a position where they have no choice but to take the last square, resulting in a win for the first player. This requires careful planning and analysis of the game state to identify the optimal moves that lead to a winning position.

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To prove that if ∠ACB is congruent to ∠ABC, then ∠XCA is congruent to ∠YBA, we can use the transitive property of congruence. In the proof above, we start with the given information that ∠ACB is congruent to ∠ABC.

By applying the transitive property of congruence, we can establish that ∠XCA is congruent to ∠ABC and ∠YBA is congruent to ∠ACB. Finally, using the transitive property once again, we conclude that ∠XCA is congruent to ∠YBA.

Statement                            | Reason

------------------------------------|---------------------------------------

1. ∠ACB ≅ ∠ABC.                       | Given

2. ∠ACB ≅ ∠ACB.                       | Reflexive property of equality

3. ∠ABC ≅ ∠ACB.                       | Symmetric property of congruence

4. ∠XCA ≅ ∠ACB.                       | Given

5. ∠XCA ≅ ∠ABC.                       | Transitive property of congruence (3, 4)

6. ∠YBA ≅ ∠ABC.                       | Given

7. ∠YBA ≅ ∠ACB.                       | Transitive property of congruence (1, 6)

8. ∠XCA ≅ ∠YBA.                       | Transitive property of congruence (5, 7)

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The final answer in scientific notation is 50.778 x 10^−29.

To simplify (6.51 x 10^−11)(7.8 x 10^−18), we can multiply the numerical parts and add the exponents of 10:

(6.51 x 10^−11)(7.8 x 10^−18) = 6.51 x 7.8 x 10^(-11 - 18) = 50.778 x 10^(-29).

Therefore, the final answer in scientific notation is 50.778 x 10^−29.

To simplify the expression (6.51 x 10^−11)(7.8 x 10^−18), we need to perform two steps:

Step 1: Multiply the numerical parts:

6.51 x 7.8 = 50.778

Step 2: Add the exponents of 10:

10^(-11) x 10^(-18) = 10^(-11 - 18) = 10^(-29)

Combining the results from Step 1 and Step 2, we get:

(6.51 x 10^−11)(7.8 x 10^−18) = 50.778 x 10^−29

The final answer is expressed in scientific notation as 50.778 x 10^−29.

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An angle bisector divides the opposite side of a triangle into segments that are proportional to the lengths of the adjacent sides.

Conjecture: When an angle of a triangle is bisected by a line segment, it divides the opposite side into two segments that are proportional to the adjacent sides of the triangle.

When an angle of a triangle is bisected by a line segment, it creates two smaller angles that are congruent. According to the Angle Bisector Theorem, the line segment divides the opposite side of the triangle into two segments.

Let's consider a triangle ABC where AD is the angle bisector of angle A, intersecting side BC at point D. According to the conjecture, we can state that:

AD/DB = AC/CB

This means that the ratio of the length of the segment AD to the length of the segment DB is equal to the ratio of the length of the side AC to the length of the side CB. In other words, the segments created by the angle bisector are proportional to the adjacent sides of the triangle.

This conjecture is based on the Angle Bisector Theorem and the concept of proportionality. It can be proven using geometric properties and algebraic methods, providing a useful tool for solving various problems involving angle bisectors and segment lengths in triangles.

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Answer:

[tex]\dfrac{\text{d}y}{\text{d}x}=3e^{3x-5}[/tex]

Step-by-step explanation:

Differentiating from First Principles is a technique to find an algebraic expression for the gradient at a particular point on the curve.

[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Differentiating from First Principles}\\\\\\$\text{f}\:'(x)=\displaystyle \lim_{h \to 0} \left[\dfrac{\text{f}(x+h)-\text{f}(x)}{(x+h)-x}\right]$\\\\\end{minipage}}[/tex]

The point (x + h, f(x + h)) is a small distance along the curve from (x, f(x)).

As h gets smaller, the distance between the two points gets smaller.

The closer the points, the closer the line joining them will be to the tangent line.

To differentiate y = e^(3x-5) using first principles, substitute f(x + h) and f(x) into the formula:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{e^{3(x+h)-5}-e^{3x-5}}{(x+h)-x}\right][/tex]

Simplify the numerator:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{e^{3x+3h-5}-e^{3x-5}}{(x+h)-x}\right][/tex]

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{e^{3x-5}e^{3h}-e^{3x-5}}{h}\right][/tex]

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{e^{3x-5}(e^{3h}-1)}{h}\right][/tex]

Apply the Product Law for Limits, which states that the limit of a product of functions equals the product of the limit of each function:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0}\left[e^{3x-5}\right] \cdot \lim_{h \to 0} \left[\dfrac{(e^{3h}-1)}{h}\right][/tex]

Since the first function does not contain h, it is not affected by the limit:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{h \to 0} \left[\dfrac{(e^{3h}-1)}{h}\right][/tex]

Transform the numerator of the second function.

[tex]\textsf{Let\;\;$e^{3h}-1=n \implies e^{3h}=n+1$}[/tex]

[tex]\textsf{You will notice that as\;\;$h \to 0, \;e^{3h} \to 1$,\;so\;\;$n \to 0$.}[/tex]

Take the natural log of both sides and rearrange to isolate h:

[tex]\ln e^{3h}=\ln(n+1)[/tex]

    [tex]3h=\ln(n+1)[/tex]

      [tex]h=\dfrac{1}{3}\ln(n+1)[/tex]

Therefore:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{n}{\frac{1}{3}\ln(n+1)}\right][/tex]

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{3n}{\ln(n+1)}\right][/tex]

Rewrite the fraction as 1 divided by the reciprocal of the fraction:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{1}{\frac{\ln(n+1)}{3n}}\right][/tex]

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{1}{\frac{1}{3n}\ln(n+1)}\right][/tex]

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{3}{\frac{1}{n}\ln(n+1)}\right][/tex]

Apply the Log Power Law:

[tex]\displaystyle \dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \lim_{n \to 0} \left[\dfrac{3}{\ln(n+1)^{\frac{1}{n}}}\right][/tex]

Apply the Quotient Law for Limits, which states that the limit of a quotient of functions equals the quotient of the limit of each function:

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \left[\dfrac{\displaystyle\lim_{n \to 0}3}{\displaystyle\lim_{n \to 0}\ln(n+1)^{\frac{1}{n}}}\right][/tex]

Therefore, the numerator is a constant:

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \left[\dfrac{3}{\displaystyle\lim_{n \to 0}\ln(n+1)^{\frac{1}{n}}}\right][/tex]

The limit of a function is the function of the limit.

Move the limit inside and take the natural log of that limit:

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \left[\dfrac{3}{\displaystyle \ln\left(\lim_{n \to 0}(n+1)^{\frac{1}{n}}\right)}\right][/tex]

The definition of e is:

[tex]\boxed{e=\lim_{n \to 0}(n+1)^{\frac{1}{n}}}[/tex]

Therefore:

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \left[\dfrac{3}{\displaystyle \ln\left(e\right)}\right][/tex]

As ln(e) = 1, then:

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot \left[\dfrac{3}{1}\right][/tex]

[tex]\dfrac{\text{d}y}{\text{d}x}=e^{3x-5}\cdot 3[/tex]

[tex]\dfrac{\text{d}y}{\text{d}x}=3e^{3x-5}[/tex]