Use the definition of the derivative to find the derivative of the function with respect to x. Show steps
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The derivative of the function y = -1/x-2 is 1/(x-2)².
Given, the function is y = -1/x-2
Differentiate the function with respect to x.
dy/dx = d/dx (-1/x-2)
the function is in the form of :
d/dx [f(x)g(x)] = f(x)d/dx((x)) + g(x)d/dx(f(x))
here d/dx [f(x)g(x)] = d/dx [(-1)(1/x-2)]
therefore, d/dx [(-1)(1/x-2)] = (-1)d/dx(1/x-2) +(1/x-2)d/dx(-1)
⇒ d/dx [(-1)(1/x-2)] = (-1)(-1)(x-2)⁻¹⁻¹ + (1/x-2)d/dx(0)
⇒ d/dx [(-1)(1/x-2)] = 1(x-2)⁻² + 0
⇒ d/dx [(-1)(1/x-2)] = 1/(x-2)²
Hence the derivative of the function is 1/(x-2)²
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Related Questions
I need help with the question I post as a photo.
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We will have the following:
*First:
[tex]3x+\frac{1}{4}-x+1\frac{1}{2}=2x+\frac{1}{4}+\frac{3}{2}[/tex][tex]=2x+\frac{7}{4}=2x+1\frac{3}{4}[/tex]So, the first one is not equivalent to the other expression.
*Second:
[tex]2(3x+1)=6x+2[/tex]So, the second one is equivalent to the other expression.
*Third:
[tex]3(x+1)-(1+x)=3x+3-1-x[/tex][tex]=2x+2[/tex]So, the third one is not equivalent to the other expression.
*Fourth:
[tex]4(x+1)-x-4=4x+4-x-4[/tex][tex]=3x[/tex]So, the fourth one is equivalente to the other expression.
*Fifth:
[tex]5.5+2.1x+3.8x-4.1=5.9x+1.4[/tex]So, the fifth one is equivalent to the other expression.
simplify the rational expression. 18x3y5 45x5y9
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8y+63x+9
[tex]5x + 17 = 82[/tex]simplify as much as possible
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To answer this question, we can follow the next steps:
1. Subtract 17 to both sides of the equation (we apply here the subtraction property of equality):
[tex]5x+17-17=82-17\Rightarrow5x+0=65\Rightarrow5x=65[/tex]2. To isolate the variable, x, in the equation, we need to divide by 5 to both sides of the equation, as follows:
[tex]\frac{5x}{5}=\frac{65}{5}\Rightarrow\frac{5}{5}=1\Rightarrow x=\frac{65}{5}\Rightarrow x=13[/tex]We can check this result if we substitute this last value into the original equation:
[tex]5x+17=82\Rightarrow5(13)+17=82\Rightarrow65+17=82\Rightarrow82=82[/tex]The result is always TRUE.
Therefore, the value for the unknown value of x is x = 13.
The sum of two numbers is 40. If 2 is added to the larger number, theresult is equal to twice the smaller number. What are the two numbers?
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We have 2 numbers. We can call them x and y, being x the smaller one.
The sum of this two numbers is 40, so we can write:
[tex]x+y=40[/tex]We know that if 2 is added to the larger number (that we name as y), the result is twice the smaller number, that would be 2x. Then, we can express this as:
[tex]y+2=2x[/tex]We can express y in function of x from the second equation and then replace it in the first equation to solve for x:
[tex]y+2=2x\Rightarrow y=2x-2[/tex][tex]\begin{gathered} x+y=40 \\ x+(2x-2)=40 \\ 3x-2=40 \\ 3x=40+2 \\ 3x=42 \\ x=\frac{42}{3} \\ x=14 \end{gathered}[/tex]Now, we can calculate y as:
[tex]\begin{gathered} y=2x-2 \\ y=2(14)-2 \\ y=28-2 \\ y=26 \end{gathered}[/tex]Answer: the two numbers are 14 and 26.
-Exponential and Logarithmic Functions- Solve the following, round the answer to the nearest hundredth.
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Answer:
x = 1.70
Explanation:
We were given that:
[tex]\begin{gathered} 5^x=15.4 \\ \text{Take the natural logarithm of both sides, we have:} \\ \ln 5^x=\ln 15.4 \\ x\cdot\ln 5=\ln 15.4 \\ \text{Divide both sides by ''ln5'', we have:} \\ x=\frac{\ln 15.4}{\ln 5}=1.699 \\ x=1.699\approx1.70 \\ x=1.70 \\ \\ \therefore x=1.70 \end{gathered}[/tex]Therefore, x = 1.70
Which is 56,900,000 in scientific notation?o 5.69 x 10⁷o 56.9 x 10⁷o 5.69 X 10⁶o 56.9 X 10⁶
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Answer:
5.69 x 10⁷
Explanation:
A number is said to be in the scientific notation when it is written as a product of a number between 1 and 10 and a power of 10.
The number 56,900,000 in scientific notation is 5.69 x 10⁷.
The correct choice is A.
Which choice is equivalent to the expression below?V-81A. 91B. AiC.D. -29E. -9SUBMIT
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Given the expression:
[tex]\sqrt[]{-81}[/tex]As we know, there is no square root for the negative numbers
But, using the complex numbers:
[tex]i=\sqrt[]{-1}[/tex]So, the given expression can be written as:
[tex]\sqrt[]{-81}=\sqrt[]{-1}\cdot\sqrt[]{81}=i\cdot9=9i[/tex]So, the answer will be option A) 9i
Suppose the mean income of firms in the industry for a year is 95 million dollars with a standard deviation of 17 million dollars. If incomes for the industry are distributed normally, what is the probability that a randomly selected firm will earn less than 112 million dollars? Round your answer to four decimal places.
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0.8413 is the probability that a random selected firm will earn less than 112 million dollar
What is probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1.
Let X be a random variable represents the income of the firm in the industry
Hence
X~ N (mean =u= 95 , standard deviation= d = 17 )
We must determine the likelihood that a randomly chosen company will make fewer than 112 million dollars in earnings ie.
P(X<112) = P(X-u/d < 112-95/17)
Z=X-u/d = 112 - 95/17 = 1
P(X<112) = P(Z-1)=0.8413
Using the standard normal probability table.
P(X<112) = 0.8413
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231231312312312312311
Answers
Answer: 3456765432345
Step-by-step explanation:
2345676543456
Explain if the triangles are similar using SAS-. If they are similar, which angles are congruent and how do you know? (Explain your reasoning using evidence like a paragraph proof NOT a rigid motion proof!)
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We have two triangles GBL and XYL.
From the picture we notice that the GL=39 and BL=34. We also notice that XL=30 and YL=27.
The SAS theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.
This means that we need that:
[tex]\frac{GL}{XL}=\frac{BL}{YL}[/tex]and that the angle between them is the same.
It is clear that the angle L is the same for both triangles, hence we only need to proof tha the sides are congruent, but in this case:
[tex]\frac{39}{30}\ne\frac{34}{27}[/tex]since the sides are not proportional, we conclude that triangles are not congruent.
pets : | bird | cat | dog | snake |frequency: | 3 | 5 | 11 | 1 |which of the following statments does NOT reflect the distribution of the data?a. one-fourth of the pets are catsb. snakes represent 10% of the pets on the farmc. the number of biirds and snakes on the farm make up 20% d. more than half of the pets on the farm are dogs
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SOLUTION:
Case: Interpreting from tables
The total number of pets is 20
Checking the options
Option A.
One-fourth of the pets are cats
[tex]\frac{1}{4}\times20=5\text{ }pets[/tex]This is TRUE from the table
Option B
Snakes represent 10% of the pets on the farm
[tex]\begin{gathered} 10\%\times20 \\ \frac{10}{100}\times20=2\text{ }pets \end{gathered}[/tex]This is FALSE from tables as there is only 1 snake
Option C
The number of biirds and snakes on the farm make up 20%
[tex]\begin{gathered} 20\%\times20 \\ \frac{20}{100}\times20=4\text{ }pets \end{gathered}[/tex]This is TRUE from the table
Option D
More than half of the pets on the farm are dogs
[tex]\frac{1}{2}\times20=10\text{ }pets[/tex]This is TRUE from the table
The Cunninghams are moving across the country. Mr.Cunningham leaves 3 hours before Mrs. Cunningham. If he averages 55 mph and sheaverages 75 mph, how many hours will it take Mrs. Cunningham to catch up to Mr. Cunninham to catch up to mr.cunningham
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Solution:
Remember, distance traveled is the rate times the time. (d = rt) Mrs. Cunningham will overtake Mr. Cunningham when they have traveled the same distance.
Mrs. Cunningham's equation will be:
[tex]d=\text{ }75t[/tex]Since he was traveling 3 hours longer, Mr. Cunningham's equation will be:
[tex]d=55(t+3)[/tex]If they travel the same distance, the equations can be set equal to each other:
[tex]\text{ }75t=55(t+3)[/tex]applying the distributive property, this is equivalent to:
[tex]\text{ }75t=55t\text{ +165}[/tex]this is equivalent to:
[tex]75t-55t\text{ = 165}[/tex]this is equivalent to:
[tex]20t\text{ = 165}[/tex]solving for t, we obtain:
[tex]t\text{ =}\frac{165}{20}=8.25[/tex]So that, we can conclude that the correct answer is:
It will take Mrs. Cunningham 8.25 hours to overtake her husband.
M/4 + q ; m=2/3 , and q= 1/4
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Given the expression:
[tex]\frac{m}{4}+q[/tex]We will find the value of the expression when m=2/3, and q= 1/4
So,
[tex]\begin{gathered} (\frac{2}{3}\div4)+\frac{1}{4} \\ \\ =(\frac{2}{3}\times\frac{1}{4})+\frac{1}{4} \\ \\ =\frac{1}{6}+\frac{1}{4}=\frac{2}{12}+\frac{3}{12}=\frac{5}{12} \end{gathered}[/tex]So, the answer will be: 5/12
A linear function has a slope of 11. Interpret this slope with a complete sentence using the words“inputs” and “outputs”. (1 point)As the inputs________,_______
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Answer
the inputs increase by 1 and the outputs increase by 11
Step-by-step explanation:
The standard form of a linear function is written as
y = mx + c
where m = slope
Since the slope is 11
y = 11x + c
This implies that the inputs increase by 1 and the outputs increase by 11
4. Ifline m has the equation y = 3x - 1, and line k is perpendicular to m and goes through the point (-4,3), find the equation of line k.
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Answer:
The equation of the line k is
[tex]y=-\frac{1}{3}x+\frac{5}{3}[/tex]Explanation:
Given that k is perpendicular to line m, defined as:
y = 3x - 1
the slope of k is the negative reciprocal of the slope of line m.
The slope of m is 3
The negative reciprocal of m is -1/3 (this is the slope of k)
Therefore, k is in the form
[tex]y=-\frac{1}{3}x+b[/tex]Since this line passes through the point (x, y) = (-4, 3), we can use this to obtain the value for the y-intercept, b
[tex]\begin{gathered} 3=-\frac{1}{3}(-4)+b \\ \\ 3=\frac{4}{3}+b \end{gathered}[/tex]Solving for b by subtracting 4/3 from both sides
[tex]\begin{gathered} b=3-\frac{4}{3} \\ \\ =\frac{5}{3} \end{gathered}[/tex]The equation is therefore,
[tex]y=-\frac{1}{3}x+\frac{5}{3}[/tex]Match each expression on the left with its sum on the right. Some answer options on the right will not be used.
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To match the expression with the sum, what you have to do is solve each sum.
Remember that to sum/subtract two fractions, both of them should be expressed using the same denominator,
1)
[tex]-\frac{2}{3}+\frac{5}{6}[/tex]The denominators of these fractions are "3" and "6", the least common denominator between both values is 6. To express the first fraction as its equivalent with denominator 6, you have to multiply it by 2:
[tex]-\frac{2\cdot2}{3\cdot2}+\frac{5}{6}=-\frac{4}{6}+\frac{5}{6}[/tex]Now you can proceed to add both fractions:
[tex]-\frac{4}{6}+\frac{5}{6}=\frac{-4+5}{6}=\frac{1}{6}[/tex]The result for this sum is 1/6
2)
[tex]\frac{7}{12}+(-\frac{3}{4})[/tex]First, simplify both symbols, when a plus symbol and a minus symbol and next to each other, the plus sign gets canceled:
[tex]\frac{7}{12}+(-\frac{3}{4})=\frac{7}{12}-\frac{3}{4}[/tex]To subtract both fractions the first step is to express them using the same denominator. The least common denominator between 12 and 4 is 12, to express -3/4 as its equivalent with denominator 12, you have to multiply the fraction by 3:
[tex]\frac{7}{12}-\frac{3\cdot3}{4\cdot3}=\frac{7}{12}-\frac{9}{12}[/tex]Next, subtract both fractions:
[tex]\frac{7}{12}-\frac{9}{12}=\frac{7-9}{12}=-\frac{2}{12}[/tex]The result is no in its simplest form, 2 and 12 are divisible by 2, so to simplify the fraction you have to divide the numerator and denominator by 2:
[tex]-\frac{2\div2}{12\div2}=-\frac{1}{6}[/tex]The result for this expression is -1/6
3)
[tex]-\frac{1}{4}+\frac{3}{8}[/tex]Same as before, the first step is to express both fractions with the same denominator. the least common denominator for both fractions is 8. To express -1/4 as its equivalent with denominator 8, you have to multiply the fraction by 2
[tex]-\frac{1\cdot2}{4\cdot2}+\frac{3}{8}=-\frac{2}{8}+\frac{3}{8}[/tex]Next, add both fractions:
[tex]-\frac{2}{8}+\frac{3}{8}=\frac{-2+3}{8}=\frac{1}{8}[/tex]The result for this sum is 1/8
So the corresponding matches are:
[tex]\begin{gathered} 1)-\frac{2}{3}+\frac{5}{6}=\frac{1}{6} \\ 2)\frac{7}{12}+(-\frac{3}{4})=-\frac{1}{6} \\ 3)-\frac{1}{4}+\frac{3}{8}=\frac{1}{8} \end{gathered}[/tex]describe and state a situation where you can apply the concepts of point, line, and plane in real-life situation
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Point
A point is an exact position or location on a plane surface.
Real life situation: The end of a knife used for cutting meat in preparation for a meal.
Description: The end of knife is a point whose location is precise. It has ability to pass through surfaces easily due to this fact.
Line
A line is a one-dimensional figure, which has length but no width.
Real-life situation : The edge of a wall or rectangular table.
Description: The edge of a table represent a distance from one end to another end.
Plane:
A plane is a flat, two-dimensional surface that extends indefinitely.
Real-life situation: The surface of the floor or table.
Description: The surface of a floor is such that it can accommodate things on it showing that it is 2-dimensional.
Mari pushed a cube- shaped box to explore force. She examined the attributes of the box. Does a face of her box have a right angle? Explain
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The face of a cuboid box have 4 right angles.
What is mean by Cuboid?
A cuboid is the solid shape or three-dimensional shape. A convex polyhedron which is bounded by six rectangular faces with eight vertices and twelve edges is called cuboid.
Given that;
Mari pushed a cube- shaped box to explore force.
And, She examined the attributes of the box.
Now,
In the cube shape, faces are all squares.
And, A square is a quadrilateral in which all angles are 90 degree.
Thus, The face of a cuboid box have 4 right angles.
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Which of the following graphs is a polynomial function with intercepts of(-2,0), (1, 0), and (4, 0)711-15 4NO C.O D.
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Explanation
We are given the following:
We are required to determine which of the following graphs is a polynomial function with intercepts of
(-2,0), (1, 0), and (4, 0).
This can be achieved by looking for the graph that crosses the x-axis at the points -2, 1 and 4.
Hence, the answer is option C.
how many 3×3 cm squares would fit in a 4×6 inch rectangle
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Answer:2
Step-by-step explanation:
6 divided by 2 would be 3, which is the length size of the square. The height does not allow to stack, which means you can fit two squares.
Find the slope of the line that passes through (4,2) and (2,1) which set up in the formula is correct? Select all that apply.
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The formula for calculating the slope of a line passing through the points (x1, y1) and (x2, y2) is expressed as:
[tex]slope=\frac{y_2-y_1}{x_2-x_1}\text{ }or\text{ }\frac{y_1-y_2}{x_1-x_2}[/tex]Given the coordinate points (4,2) and (2,1), the possible set up formulas are:
[tex]\begin{gathered} x_1=4 \\ y_1=2 \\ x_2=2 \\ y_2=1 \end{gathered}[/tex][tex]\begin{gathered} slope=\frac{1-2}{2-4} \\ slope=\frac{2-1}{4-2} \end{gathered}[/tex]This are the slopes of the line formula
A park meadow is planted with wildflowers. The Parks Department plans to extend the length of the rectangular meadow by x meters. Which expressions represent the total area, in square meters, after the meadow's length is increased? Select all that apply. 15. A 310 + x B 15.5(20x) C 20x + 15.5 D 15.5x + 310 E 15.5(20 + x) F 35.5 + x Ilse the distributi
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We have the following:
The area would be the length by the width, but since x amount was added to the length, it would be like this
[tex]\begin{gathered} A=w\cdot l \\ w=15.5 \\ l=20+x \end{gathered}[/tex]replacing
[tex]A=15.5\cdot(20+x)=310+15.5x[/tex]Therefore, the answer is E and D
Find angle a in the taper shown,x = 9.342 inchesy = 6.692 inchesz = 2.952 inches
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We need to find angle a in the figure.
We know that:
x = 9.342 inches
y = 6.692 inches
z = 2.952 inches
We can do so by finding the legs in the following triangle:
The adjacent leg is x. And the opposite leg is found by subtracting z from y, and then dividing the result by two (assuming the figure is symmetric):
[tex]\frac{y-z}{2}[/tex]Thus, we have:
[tex]\begin{gathered} \sin a=\frac{\text{ opposite leg}}{\text{ adjacent leg}} \\ \\ \sin a=\frac{\frac{y-z}{2}}{x} \\ \\ \sin a=\frac{\frac{6.692-2.952}{2}}{9.342} \\ \\ \sin a=\frac{1.87}{9.342} \\ \\ a=\arcsin\left(\frac{1.87}{9.342}\right) \\ \\ a\cong11.55\degree \end{gathered}[/tex]Gina left home, riding her bicycle at a rate of 25 miles per hour. Sean left 1 hour later, riding at a rate of 30 miles per hour. How long will it take Sean to catch up to Gina?
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As per the distance formula, it take 1 hour of time for Sean to catch up to Gina.
Distance formula:
The equation that relates the distance, rate, and time is
d = rt
Where d represents the distance traveled, r represents the rate, and t represents the time.
Given,
Gina left home, riding her bicycle at a rate of 25 miles per hour. Sean left 1 hour later, riding at a rate of 30 miles per hour.
Here we need to find the time take by Sean to catch up Gina.
Let us consider x be the time when Gina left the home.
Then, Sean left 1 hour later from her time.
So, it can be written as,
=> x + 1
As the Distance traveled is the same, the ratio of Speed in case 1 to the Speed in case 2 will be the inverse of the Time taken in both cases.
Therefore, the ratio of Speed in both cases
=> 25 : 30
=> 25/30
=> 5/6
Therefore, it can be written as,
x/x+1 = 5/6
When we cross multiply them, then we get,
5x + 5 = 6x
x = 5.
If Gina left at the time of 5, then Sean left at the time of 6.
So, it take 1 hour for Sean to catch up to Gina.
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what is 40+56 in GCF
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The GCF stands for greatest common factor. To represent a sum by its GCF we need to use the distributive property and we need to first find the GCF of the numbers. Let's break each number by its factors:
[tex]\begin{gathered} 40=2\cdot2\cdot2\cdot5 \\ 56=2\cdot2\cdot2\cdot7 \end{gathered}[/tex]We now multiply the numbers that appear on both.
[tex]\text{GCF}=2\cdot2\cdot2=8[/tex]We now apply the distributive property:
[tex]8\cdot(5+7)[/tex]numbers in order from greatest to least 1/5 0.12 0.17
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1/5 = 0.2
the order is:
1/5
0.17
0.12
Solve the system using the elimination method. State your final answer as an ordered pair. DO NOT include any spaces in your answers.
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Given:-
let
5x-4y=1 be the equation 1
-5x-10y=-15 be the equation 2
step 1-
add equation 1 and 2
we get=
-14y=-14
y=1
this is required value of y
we are going to put this value of y in equation 1
we get
5x-4(1)=1
5x-4=1
5x=1+4
5x=5
x=1
this is required value of x
hence value of x and y are(1,1)
Line Graph: This time you will not have the numbers on the x and y axis. You will need to decide which number to use (1, 2, 3... or 2,4,5.... Or 5, 10, 15...) 3: Creating Graphs Create a single line graph using the following table. Time goes on the x axis Rainfall goes on the y axis Make sure to do the following: Label the x and y axis Create a title 10 15 20 Time (minutes) 25 30 35 40 25 55 45 60 50 35 40 Speed (of car) (km/min)
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Line Graph:
A line graph is used to show how the data points are changing with respect to time.
For Example:
A line graph may be used to show the average rainfall over the entire month.
For the given scenario we have,
X-axis = Time in minutes
Y-axis = Speed of car in km/min
Title of graph = Speed of Car Vs Time
Data points for time = 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
Data points for speed = 25, 30, 35, 40, 25, 55, 45, 60 50 35 40
This is how the line graph looks like.
It is showing the speed of the car in km/min over an interval of 60 minutes in steps of 5 minutes.n steps of
Procedure:
• Draw and label the x-axis and y-axis.
,• Label the data points on both axis.
,• Draw the data points.
,• Join the data points with a line.
,• We are done.
,•
Multiplying and Dividing Integers 10-16 Name: 1. As a cold front passed through Temple, the temperature changed steadily over 6 hours. Altogether it change -18 degrees. What was the change in temperature each hour for the 6 hours? a.-18 - 6 = -3 degrees b. 18 - 6 = 3 degrees c. 18 + 6 = 24 degrees d. 18 - 6 = 12 degrees 2. Q. Four college roommates rented an apartment together. When they moved out, they were charged $1500 for damages to the carpet and walls. The roommates agreed to equally share the cost. What integer represents how much each person will have to pay?
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Given the total change in temperature in 6 hours, it is necessary to divide it by the number of hours
[tex]-\frac{18}{6}=-3[/tex]The change in temperature each hour is -3 degrees
urgently need help with question 30, it’s Venn diagram, is it valid or not valid & is the argument sound or not?
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For statement 30:
Premise: All fruits are foods with sugar;
Premise: Chocolate bars contain sugar;
Conclusion: Chocolate bars are fruit.
Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether chocolate is fruit.