Answer:
x+2y=12-------(1)
x-2y=3---------(2)
Adding equations 1 and 2
we get
2x=18
x=9
Equation 1
9+2y=15
2y=15-9
2y=6
y=3
The solution of the given system is x=9, y=3
Step-by-step explanation
Dan's mean average on 5 exams is 86 determine the sum of his score
Answer: 430
Step-by-step explanation:
An average of 5 scores can be found via: (the sum of the scores)*5. Thus, simply multiply 86*5 to get that the sum of his scores is 430
Hope it helps <3
The smaller of two numbers is one-half the larger, and their sum is 27. Find the numbers. Answer: The numbers are ___ ___ ___
Answer:
the smaller is 9 while the digger is 18
Two types of survey questions are open questions and closed questions. An open question allows for any kind of response; a closed question allows for only a fixed response. An open question and a closed question with its possible choices are given below. List the advantages and disadvantages of each question.
Open question: What can be done to get students to eat healthier foods?
Closed question: How would you get students to eat healthier foods?
1. Mandatory nutrition course
2. Offer only healthy foods in the cafeteria and remove unhealthy foods
3. Offer more healthy foods in the cafeteria and raise the prices on unhealthy foods
What are the advantag?
A. It is easy to compare the results of surveys with open questions.
B. An open question allows for new solutions to be introduced.
C. It is easy to quantify the responses of open questions.
D. An open question allows the respondent to go in-depth with their answer.
What are the disadvantages of an open question?
A. It is difficult to compare the results of surveys with open questions.
B. It is difficult to quantify the responses of open questions.
C. The form of the question may influence the opinion of the respondent.
D. An open question limits the possible responses of the respondent.
Answer and explanation:
Advantages of open questions:
B. An open question allows for new solutions to be introduced.
D. An open question allows the respondent to go in-depth with their answer.
Disadvantages of open questions:
A. It is difficult to compare the results of surveys with open questions.
B. It is difficult to quantify the responses of open questions.
Advantages of closed questions:
- It is easy to compare the results of surveys with closed questions.
- It is easy to quantify the responses of closed questions.
Disadvantages of closed questions:
- A closed question limits the possible responses of the respondent.
-The researcher must spend time to generate a satisfactory list of possible responses from the respondent.
- possible answers may also be insufficient to be useful.
Open questions and closed questions in research questionnaires are two popular methods of getting data from respondents. They both have advantages and disadvantages and can be used interchangeably or together in questionnaires.
Closed Questions have defined answer alternatives or answer categories. Eg: Which is your favourite time of the day? (Morning or Evening etc)
Open Questions have free answers, not clear defined alternatives or categories. Eg : Why do you like morning/ evening time the most ?
Closed Questions give brief concrete answers, Open Questions have more essence of subjective & personal elaboration.
Advantages of Open Questions
Open questions allows for new solutions to be introduced. Open questions allows the respondent to go in-depth with their answer.Disadvantages of Open Questions
It is difficult to compare the results of surveys with open questions.It is difficult to quantify the responses of open questions.Advantages of Closed Questions
It is easy to quantify the responses, with these questions. It is easy to compare the results, with these questionsDisadvantages of Closed Questions
The form of question may influence potential of respondent. It limits the possible responses of respondentTo learn more, refer https://brainly.com/question/2855510?referrer=searchResults
I reallly need help with this
Answer:
m<5 == m<1 since alternate interior angles are same value
m<3 == m<6 since alternate interior angles are same value
m<6 + m<5 + m<2 = m<1 + m<2 + m<3 = 180
Step-by-step explanation:
The use of alternate interior angles definition allows for you to make this completion. You can use this since you have a line intersecting a point on two parallel lines. From here, you know that the measures of the angles are the same as the measure of the line, thus you have proven the internal sum of angles to be 180 degrees.
What are the explicit equation and domain for a geometric sequence with a first term of 2 and a second term of −8?
Step-by-step explanation:
the common ratio is -4
Gn=2 (-4)^n-1
Of the following points, name all that lie on the same horizontal line? (1,-2) (-1,2) (-2,1) (2,-1)
Answer:
None
Step-by-step explanation:
Let’s graph the points first.
Look at the image below.
And by looking at the image we can tell that none of the points lay on the same horizontal line.
Thus,
none of the points lay on the same horizontal line.
Hope this helps :)
Select the correct text in the table. Use the fundamental theorem of algebra to determine whether each statement is sometimes true, always true, or never true.
1. A quadratic function has 2 distinct roots. always sometimes never
2. A cubic function has at least 1 real root. always sometimes never
3. A function with a degree of 5 has 5 roots. always sometimes never
4. A quadratic function can have only 1 complex solution. always sometimes never
Answer:
1. Sometimes
2. Sometimes
3. Always
4. Sometimes
Step-by-step explanation:
1. Quadratic function : in which maximum power of [tex]x[/tex] is two.
The roots of quadratic function can be either equal or different.
For example:
[tex]x^{2} -2x+1[/tex] will have two equal roots i.e. 1 and 1.[tex]x^{2} -3x+2[/tex] will have two different roots i.e. 1 and 2.So, sometimes is the correct answer.
2. Cubic function has atleast 1 real root.
Cubic function has maximum power of [tex]x[/tex] as 3.
If the coefficients are real numbers then atleast 1 real root.
If the coefficients are imaginary in nature, then this is not true.
For example:
Cubic equation [tex]x^3 +i = 0[/tex] does not have any real root.
Cubic equation [tex]x^3 +1 = 0[/tex] has a real root x = -1.
So, it is sometimes true.
3. A function with degree 5 i.e. maximum power of [tex]x[/tex] as 5 will have 5 roots.
It is always true that a function will have number of roots equal to its degree.
4. Quadratic function can have only 1 complex solution.
Two complex solutions are also possible for a quadratic function.
For example:
[tex]x^{2} +1=0[/tex] will have two imaginary roots: [tex]x=i, -i[/tex]
It is also possible to have 1 complex solution,
For example:
[tex](x-1)(x-i) = 0[/tex] will have one complex root and one real root.
So, the statement is sometimes true.
Answer:
MY ANSWER IS CORRECT IN PLATO!!!
1. Sometimes
2. Always
3. Always
4. Never
Step-by-step explanation:
1. A quadratic function has 2 distinct roots SOMETIMES
2. A cubic Function has at least 1 root ALWAYS
3. A function with a degree of 5 has 5 roots ALWAYS
4. A quadratic function can have only 1 complex solution NEVER
I JUST GOT 100% on the quiz in PLATO
Given the diagram below, what is cos(45*)?
A.
B.
C.
D.
Answer:
The answer is option B
Step-by-step explanation:
To find cos 45° we must first find the adjacent and the hypotenuse
Let the adjacent be x
Let the hypotenuse be h
To find the adjacent we use tan
tan ∅ = opposite / adjacent
From the question
the opposite is 9
So we have
tan 45 = 9 / x
x tan 45 = 9
but tan 45 = 1
x = 9
Since we have the adjacent we use Pythagoras theorem to find the hypotenuse
That's
h² = 9² + 9²
h² = 81 + 81
h² = 162
h = √162
h = 9√2
Now use the formula for cosine
cos∅ = adjacent / hypotenuse
The adjacent is 9
The hypotenuse is 9√2
So we have
cos 45 = 9/9√2
We have the final answer as
cos 45 = 1 / √2Hope this helps you
Andreas wants to estimate how many fish there are in a lake. One day he catches 20 fish,marks them,and returns them to the lake. The next day he catches 10 fish of which four are marked. Estimate the number of fish in the lake.
Answer:
50
Step-by-step explanation:
The ratio of marked fish to caught fish is 4:10, we need to solve for x in 20:x, since 4 * 5 = 20, 10 * 5 = x so x = 50 fish.
Aiden is trying to pick up some lawn mowing jobs over the weekend to make extra money for a school trip. Each lawn in his neighborhood takes an average of 40 minutes to mow, and Aiden has no more than 11 hours, or 660 minutes, of available time to mow lawns. If Aiden mows his grandparents' farm which takes him 110 minutes, and x represents the number of lawns he mows in his neighborhood, which inequality represents this situation?
A.
40x + 110 ≤ 660
B.
110x + 40 ≤ 660
C.
110x + 40 ≥ 660
D.
40x + 110 ≥ 660
Answer:
A.
Step-by-step explanation:
40x is the number of lawns he can do, less the time to do his grandparents time (added to other law time) and he has 660 mins of less to complete them.
Answer:
a. 40x + 110 ≤ 660
Step-by-step explanation:
If John has pairs of red, orange, yellow, blue and green socks, how many can he wear them in over 5 days, repetition is allowed because hog is alright with wearing dirty socks. A)5! B)25 C)5^5 D)125
Answer:
Total ways = 5×5×5×5×5
Total ways = 5^5
Total ways = 3125
Therefore, the correct option is C) 5^5
Step-by-step explanation:
John has pairs of red, orange, yellow, blue and green socks.
Which means that John has 5 different colors pairs of socks.
We are asked to find out in how many ways can he wear them over 5 days.
1st Day:
On the first day John has 5 ways to choose from.
2nd Day:
On the second day John has 5×5 ways to choose from.
(Since repetition is allowed)
3rd Day:
On the third day John has 5×5×5 ways to choose from.
4th Day:
On the fourth day John has 5×5×5×5 ways to choose from.
5th Day:
On the fifth day John has 5×5×5×5×5 ways to choose from.
Total ways = 5×5×5×5×5
Total ways = 5^5
Total ways = 3125
Therefore, the correct option is C) 5^5
Given f(x) and g(x) = f(x) + k, use the graph to determine the value of k. A) 2 B) 3 C) 4 D) 5 IF YOU ANSWER IN 5 MINUTES I WILL GIVE BRAINLIEST!!!!!!!!!!!!!!!!!!!!!!
Ans k = 4
Step-by-step explanation:
Here g(x) =[tex]\frac{-1}{3}x + 1[/tex] and
f(x) = [tex]\frac{-1}{3} x -3[/tex]
Now, g(x) = f(x) + k
or, [tex]\frac{-1}{3}x + 1[/tex] = [tex]\frac{-1}{3} x -3 + k[/tex]
or, 1 + 3 = k
So, k = 4 Answer.
please show on graph (with x and y coordinates) state where the function x^4-36x^2 is non-negative, increasing, concave up
Answer:
[tex] y'' =12x^2 -72=0[/tex]
And solving we got:
[tex] x=\pm \sqrt{\frac{72}{12}} =\pm \sqrt{6}[/tex]
We can find the sings of the second derivate on the following intervals:
[tex] (-\infty<x< -\sqrt{6}) , y'' = +[/tex] Concave up
[tex]x=-\sqrt{6}, y =-180[/tex] inflection point
[tex] (-\sqrt{6} <x< \sqrt{6}), y'' =-[/tex] Concave down
[tex]x=\sqrt{6}, y=-180[/tex] inflection point
[tex] (\sqrt{6}<x< \infty) , y'' = +[/tex] Concave up
Step-by-step explanation:
For this case we have the following function:
[tex] y= x^4 -36x^2[/tex]
We can find the first derivate and we got:
[tex] y' = 4x^3 -72x[/tex]
In order to find the concavity we can find the second derivate and we got:
[tex] y'' = 12x^2 -72[/tex]
We can set up this derivate equal to 0 and we got:
[tex] y'' =12x^2 -72=0[/tex]
And solving we got:
[tex] x=\pm \sqrt{\frac{72}{12}} =\pm \sqrt{6}[/tex]
We can find the sings of the second derivate on the following intervals:
[tex] (-\infty<x< -\sqrt{6}) , y'' = +[/tex] Concave up
[tex]x=-\sqrt{6}, y =-180[/tex] inflection point
[tex] (-\sqrt{6} <x< \sqrt{6}), y'' =-[/tex] Concave down
[tex]x=\sqrt{6}, y=-180[/tex] inflection point
[tex] (\sqrt{6}<x< \infty) , y'' = +[/tex] Concave up
Taylor and Jeff have been selling frozen pizzas for a class fundraiser. Taylor has sold half as many
pizzas as Jeff. Together they have sold a total of 126 pizzas. How many pizzas did Taylor sell?
Answer:
Taylor sold 42 pizzas
Step-by-step explanation:
Make a system of equations where t represents the number of pizzas Taylor sold and j represents the number that Jeff sold:
t + j = 126
j = 2t
We can solve this system by substitution, since we can substitute 2t as j.
t + j = 126
t + 2t = 126
3t = 126
t = 42
Taylor sold 42 pizzas.
Answer:
Step-by-step explanation:
Let x represent the number of pizzas that Tailor sold.
Let y represent the number of pizzas that Jeff sold.
Together they have sold a total of 126 pizzas. This means that
x + y = 126- - - - - - - - -1
Taylor has sold half as many
pizzas as Jeff. This means that
x = 1/2 × y = y/2
Substituting x = y/2 into equation 1, it becomes
y/2 + y = 126
Multiplying both sides of the equation by 2, it becomes
y + 2y = 252
3y = 252
y = 252/3
y = 84
x = y/2 = 84/2
x = 42
Taylor sold 42 pizzas
Jacob needs to know if the volume of a storage bin is under 3,000 cubic feet. The
dimensions of the bin are 17 ft. X 15 ft. x 10 ft.
a. Is the bin under 3,000 cubic ft.?
b. If yes, by how much?
Answer:
It is less than 3000 ft^3 by 450 ft^3
Step-by-step explanation:
The volume of the bin
V = l*w*h
V = 17*15*10
V =2550 ft^3
If it less than 3000 ft^3
V = 3000- 2550 =450 ft^3
If is less by 450 ft^3
Answer:
Let’s first multiply all the numbers given
Since it wants the volume we need to use the formula
LxWxH
17x15x10=2,550
Part A: yes the bin is under 3,000
Part B: by 450 more because if you subtract 3,000 and 2,550 you will get 450
Hope this helps! :)
Which equation shows function g in factored form?
g(x) = 2x^2 – 6x – 56
O A. g(x) = 2(x-4)(x + 7)
OB.g(x) = 2(x2-3x-28)
OC. g(x) = (2x + 7)(x-8)
OD. g(x) = 2(x + 4)(x - 7)
Answer:
[tex]\boxed{\sf Option \ D}[/tex]
Step-by-step explanation:
[tex]\sf g(x) = 2x^2-6x-56\\[/tex]
Factorizing using mid term break formula
[tex]\sf g(x) = 2x^2-14x+8x-56\\g(x) = 2x(x-7)+8(x-7)\\g(x) = (2x+8)(x-7)\\g(x) = 2(x+4)(x-7)[/tex]
Someone help me please
Answer:
31 m
Step-by-step explanation:
v=l*w*h since it is a cube then all sides (a) are equal:
v=(a*a*a)=a^3
v1+v2=1331 for the first two boxes(
a³+a³=∛1331
l=w=h=11*2=22
v=729 ( for the second cube)
a=∛729=9
9+22=31 m
Nigel is going from London, UK, to Moscow, Russia, by train. He goes 517 kilometers
on a train from London to Paris, France. He takes another train 837 kilometer to Munich
Germany, and switches trains in Munich to ride 399 kilometers on a train to Vienna,
Austria. His last train ride carries him 2,088 kilometers from Vienna to Moscow
Find his total distance by first rounding each distance to the nearest hundred kilometers
before adding
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
1) List out all distances singly, before rounding off.
i) From London to Paris → 517Km
ii) From Paris to Munich → 837Km
iii) From Munich to Vienna → 399Km
iv) From Vienna to Moscow → 2,088Km
2) Estimating the distances singly on the basis of rounding off to the nearest 100.
i) 517Km ≈ 500Km to the nearest 100
ii) 837Km ≈ 800Km to the nearest 100
iii) 399Km ≈ 400Km to the nearest 100
iv) 2,088Km ≈ 2,100Km to the nearest 100
3) Summing all estimated distances.
500 + 800 + 400 + 2100
= 3,800Km.
NB: when a number is equal or above 50, you are to round it off as 100; when the number is below 50, you are to round off as zero (0).
20 points please help!!!
Answer:
a = 16
b = [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
Length of the design 16 inches is represented by the point (0, 16) and length of 12 inches by (1, 12).
That means these points lie on the graph of the function 'f' represented by,
f(x) = a(b)ˣ
For the point (0, 16),
f(0) = a(b)⁰
16 = a(1)
a = 16
For another point (1, 12),
f(1) = a(b)¹
12 = ab
12 = 16(b) [Since a = 16]
b = [tex]\frac{12}{16}[/tex]
b = [tex]\frac{3}{4}[/tex]
Therefore, values of a and b are 16 and [tex]\frac{3}{4}[/tex] respectively.
Which point satisfies the system of equations y = 3x − 2 and y = -2x + 3? PLEASE I NEED IT NOW
Answer:
(1,1)
Step-by-step explanation:
y = 3x − 2
y = -2x + 3
the point that satisfies the system:
when the two functions are equal
3x-2=-2x+3
3x+2x=3+2
5x=5
x=5/5=1
when x=1
y=-2x+3
y=-2(1)+3
y=1
Answer:
(1,1) or x=1 and y=1
Step-by-step explanation:
Since both equations are equal to y you can use substitution, replacing he y in y=3x-2 with -2x+3, giving you the equation -2x+3=3x-2. We can use this to solve for x. To isolate the variable you first need to add 2 to both sides, giving you -2x+5=3x, then add 2x to both sides. This gives you 5=5x, lastly you need to divide both sides by 5 to get x=1. Now you can plug x(1) into 1 of the 2 given equations to find y, I'll use y=3x-2. When x is plugged in as 1 you get the equation y=3(1)-2, after multiplying you get y=3-2, then you subtract to get y=1. In order to make sure this is the correct solution you can also plug in x as 1 and now y as 1 into the other equation, y=-2x+3. We can plug 1 in for x and solve for -2(1)+3 to get -2+3, or 1, so y is 1 for both equations, therefore the solution we got was correct, x=1 and y=1, or (1,1).
THe graph is going further than the outline ben 10 benden
Answer:
EB = 9
Step-by-step explanation:
CD = AB
The line with the value of five that also forms a right angle with EB is a perpendicular bisector to AB.
So the value of EB is half of AB (AB is equal to CD).
18/2 = 9
how do you graph X+2y=6
Answer:
x + 2y = 6
2y = -x + 6
y = -1/2x + 3
So, you will have a downward sloping, less steep line with an intercept at (0, 3).
You can use the Math is Fun Function Grapher and Calculator to graph the line.
Hope this helps!
A landscaping company charges $50 per cubic yard of mulch plus a delivery charge of $24. Find a
linear function which computes the total cost C(in dollars) to deliver a cubic yards of mulch.
C(x) =
Answer: c(x) = $50*x + $24
Step-by-step explanation:
First, this situation can be modeled with a linear equation like:
c(x) = s*x + b
where c is the cost, s is the slope, x is the number of cubic yards of mulch bought, and b is the y-intercept ( a constant that no depends on the number x)
Then we know that:
The company charges $50 per cubic yard, so the slope is $50
A delivery charge of $24, this delivery charge does not depend on x, so this is the y-intercept.
Then our equation is:
c(x) = $50*x + $24
This is:
"The cost of buying x cubic yards of mulch"
Which of the following is equivalent to4−(−5∗9−1)÷2+(5)2−7?
Answer:
-20
Step-by-step explanation:
Follow the PEDMAS order (from top to bottom):
Parentheses
Exponents
Division and Multiplication
Addition and Subtraction
(-5 × 9 - 1) ÷ 2 + (5)2 - 7
(-45 - 1) ÷ 2 + 10 - 7
-46 ÷ 2 + 10 - 7
-23 + 10 - 7
-13 - 7
-20
Answer:
-20
Step-by-step explanation:
=> [tex](-5 * 9-1)/2+(5)2-7[/tex]
Expanding parenthesis
=> [tex](-45-1)/2+10-7[/tex]
=> [tex]-46/2 + 3[/tex]
=> -23 + 3
=> -20
There were females and males present at the high school pep rally. Find the ratio of males to the total number of people present. Express as a simplified ratio.
Answer: 4:9
Step-by-step explanation:
The complete question is provide in the attachment below.
Given, Number of females = 125
Number of males = 100
Total people = 120+100=225
Now, the ratio of males to the total number of people present = [tex]\dfrac{\text{Total number of males}}{\text{Total people}}[/tex]
[tex]=\dfrac{100}{225}[/tex]
Divide numerator and denominator by 25 , we get
Ratio of males to the total number of people present =[tex]\dfrac{4}{9}[/tex]
Hence, the ratio of males to the total number of people present = 4:9
I need help I just don't understand
Answer: 9/7 or -5/2
Step-by-step explanation:
We can only factorise quadratics if they're in the format ax^2 + bx + c
Re-arranging the equation gives 14x^2 + 17x - 45 = 0
Factorising this quadratic gives:
(7x - 9)(2x + 5) = 0
There are numerous ways to factorise quadratics, using a calculator or via alternate methods you may have learnt in class. (E.g. 2 numbers multiply to make (14 * -45) and add up to make (17).
This gives us our solutions.
x = 9/7 or x = -5/2
Answer:
See below.
Step-by-step explanation:
First, move all the terms to one side so we have only a zero on the right:
[tex]6x^2-17x+13=20x^2-32\\-14x^2-17x+45=0\\14x^2+17x-45=0[/tex]
(I divided everything by negative 1 in the third step. This is optional, but I like having the first term positive.)
Now, we just need to factor it. To factor, what you want to do is find two numbers a and b such that:
When a and b is multiplied together, they equal the first coefficient and constant multiplied together.
And when a and b is added together, they equal the second term.
In other words, we want to find two numbers that when multiplied equals 14(-45)=-630 and when added equals 17. Then, we can substitute this into the 17. You do this by guessing and checking. It's useful to have a calculator.
After a bit, you can find that 35 and -18 works. Thus:
[tex]14x^2+17x-45=0\\14x^2+35x-18x-45=0\\7x(2x+5)-9(2x+5)=0\\(7x-9)(2x+5)=0[/tex]
Now, the finish the problem, we just need to use the Zero Product Property and solve for x:
[tex]2x+5=0\\x=-5/2\\\\7x-9=0\\x=9/7[/tex]
Note: This only works for quadratics.
Rent for a 3 bedroom apartment is regularly $936 per month. Apartment management is now offering one month free for a 12 month lease. If you sign a one year lease and apply the free month equally across months, how much is your monthly lease amount?
Answer:
$858
Step-by-step explanation:
Monthly lease amount is ⇒ $936
Considering one moth free, yearly rental would be
⇒ 11*$936= $10296The monthly lease amount :
⇒ $10296 / 12 = $858
(09.01 MC)
In circle A shown below, Segment BD is a diameter and the measure of Arc CB is 54°:
Points B, C, D lie on Circle A. Line segment BD is the diameter of circle A. Measure of arc CB is 54 degrees.
What is the measure of ∠DBC? (4 points)
Answer: 63°
Step-by-step explanation:
Since BD is the diameter, then arc BCD = 180°
Given that arc BC = 54°, then arc CD = 180° - 54° = 126°
∠DBC is an intercepted angle -> DBC = half of arc CD
[tex]\angle DBC=\bigg(\dfrac{1}{2}\bigg)126^o\quad =\large\boxed{63^o}[/tex]
Answer:
63
Step-by-step explanation:
WILL MARK BRAINLIST----- A particular map shows a scale of 1 cm:5 km. What would the map distance be (in cm) if the actual distance to be represented is 14 km?
Answer:
2.8 cm
Step-by-step explanation:
The map scale is 1 cm : 5 km. That means that 1 cm is equal to 5 km.
To find the map distance (in cm), we have to set up a ratio.
[tex]\frac{1 cm}{5 km} = \frac{x}{14 km}[/tex]
X (the map distance in cm) is over the actual distance of 14 km.
Now cross multiply and divide.
[tex]5x = 14[/tex]
[tex]\frac{5x}{5} = \frac{14}{5}[/tex]
[tex]x = 2.8 cm[/tex]
If the actual distance to be represented is 14 km, the map distance (in cm) will be 2.8 cm.
Hope that helps.
Differentiate with respect to x and simplify your answer. Show all the appropriate steps? 1.e^-2xlog(ln x)^3 2.e^-2x(log(ln x))^3 3.sin(xe^x)^3 4.sin^3(xe^x) 5.ln(xy)=e^2y
(1) I assume "log" on its own refers to the base-10 logarithm.
[tex]\left(e^{-2x}\log(\ln x)^3\right)'=\left(e^{-2x}\right)'\log(\ln x)^3+e^{-2x}\left(\log(\ln x)^3\right)'[/tex]
[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{e^{-2x}}{\ln10(\ln x)^3}\left((\ln x)^3\right)'[/tex]
[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}(\ln x)^2}{\ln10(\ln x)^3}\left(\ln x\right)'[/tex]
[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}(\ln x)^2}{\ln10\,x(\ln x)^3}[/tex]
[tex]=-2e^{-2x}\log(\ln x)^3+\dfrac{3e^{-2x}}{\ln10\,x\ln x}[/tex]
Note that writing [tex]\log(\ln x)^3=3\log(\ln x)[/tex] is one way to avoid using the power rule.
(2)
[tex]\left(e^{-2x}(\log(\ln x))^3\right)'=(e^{-2x})'(\log(\ln x))^3+e^{-2x}\left(\log(\ln x))^3\right)'[/tex]
[tex]=-2e^{-2x}(\log(\ln x))^3+3e^{-2x}(\log(\ln x))^2(\log(\ln x))'[/tex]
[tex]=-2e^{-2x}(\log(\ln x))^3+3e^{-2x}(\log(\ln x))^2\dfrac{(\ln x)'}{\ln10\,\ln x}[/tex]
[tex]=-2e^{-2x}(\log(\ln x))^3+\dfrac{3e^{-2x}(\log(\ln x))^2}{\ln10\,x\ln x}[/tex]
(3)
[tex]\left(\sin(xe^x)^3\right)'=\left(\sin(x^3e^{3x})\right)'=\cos(x^3e^{3x}(x^3e^{3x})'[/tex]
[tex]=\cos(x^3e^{3x})((x^3)'e^{3x}+x^3(e^{3x})')[/tex]
[tex]=\cos(x^3e^{3x})(3x^2e^{3x}+3x^3e^{3x})[/tex]
[tex]=3x^2e^{3x}(1+x)\cos(x^3e^{3x})[/tex]
(4)
[tex]\left(\sin^3(xe^x)\right)'=3\sin^2(xe^x)\left(\sin(xe^x)\right)'[/tex]
[tex]=3\sin^2(xe^x)\cos(xe^x)(xe^x)'[/tex]
[tex]=3\sin^2(xe^x)\cos(xe^x)(x'e^x+x(e^x)')[/tex]
[tex]=3\sin^2(xe^x)\cos(xe^x)(e^x+xe^x)[/tex]
[tex]=3e^x(1+x)\sin^2(xe^x)\cos(xe^x)[/tex]
(5) Use implicit differentiation here.
[tex](\ln(xy))'=(e^{2y})'[/tex]
[tex]\dfrac{(xy)'}{xy}=2e^{2y}y'[/tex]
[tex]\dfrac{x'y+xy'}{xy}=2e^{2y}y'[/tex]
[tex]y+xy'=2xye^{2y}y'[/tex]
[tex]y=(2xye^{2y}-x)y'[/tex]
[tex]y'=\dfrac y{2xye^{2y}-x}[/tex]