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Please evaluate the Derivative using the Limit Process. Show all workings.
Answers
Answer:
[tex]\displaystyle f'(x)=-\frac{2}{x^{{}^{3}\!/\!{}_{2}}}[/tex]
Step-by-step explanation:
We have the function:
[tex]\displaystyle f(x)=\frac{4}{\sqrt x}[/tex]
And we want to find the derivative using the limit process.
Recall that the definition of a derivative is:
[tex]\displaystyle \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
Therefore, by substitution:
[tex]\displaystyle \lim_{h \to 0}\frac{\dfrac{4}{\sqrt{x+h}}-\dfrac{4}{\sqrt x}}{h}[/tex]
First and foremost, we can move the constant factor outside of the limit:
[tex]\displaystyle =\lim_{h \to 0}\frac{4\left(\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt x}\right)}{h}\\ \\=4\lim_{h \to 0}\frac{\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt x}}{h}[/tex]
Next, we can multiply everything by (√(x + h)(√x) to eliminate the fractions in the denominator. Therefore:
[tex]\displaystyle =4\lim_{h \to 0}\frac{\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt x}}{h}\left(\frac{\sqrt{x+h}\sqrt x}{\sqrt{x+h}\sqrt x}\right)[/tex]
Distribute:
[tex]\displaystyle =4\lim_{h \to 0}\frac{\left({\sqrt{x+h}\sqrt x}\right)\dfrac{1}{\sqrt{x+h}}-(\sqrt{x+h}\sqrt x)\dfrac{1}{\sqrt x}}{h({\sqrt{x+h}\sqrt x})}[/tex]
Distribute and simplify:
[tex]\displaystyle =4 \lim_{h\to 0}\frac{\sqrt x-\sqrt{x+h}}{h(\sqrt{x+h}\sqrt{x}) }[/tex]
Next, we can multiply both sides by the conjugate of the numerator. In other words, multiply by (√x + √(x + h)). Thus:
[tex]\displaystyle = 4\lim_{h\to 0}\frac{\sqrt x-\sqrt{x+h}}{h(\sqrt{x+h}\sqrt{x}) }\left(\frac{\sqrt x +\sqrt{x+h}}{\sqrt x +\sqrt{x+h}\right)}[/tex]
Simplify:
[tex]\displaystyle =4 \lim_{h \to 0} \frac{x-(x+h)}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}\\ \\ \\ =4 \lim_{h \to 0} \frac{x-x-h}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})} \\ \\ \\=4 \lim_{h \to 0} \frac{-h}{h(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}[/tex]
Cancel like terms:
[tex]\displaystyle =4 \lim_{h \to 0} -\frac{1}{(\sqrt{x+h}\sqrt x)(\sqrt x+\sqrt{x+h})}[/tex]
Now, we can use direct substitution. Hence:
[tex]\displaystyle \Rightarrow4 \left( -\frac{1}{(\sqrt{x+0}\sqrt x)(\sqrt x+\sqrt{x+0})}\right)[/tex]
Simplify:
[tex]\displaystyle =4\left( -\frac{1}{(\sqrt{x}\sqrt x)(\sqrt x+\sqrt{x})}\right) \\ \\ \\ =4\left( -\frac{1}{(x)(2\sqrt{x})}\right)[/tex]
Multiply:
[tex]\displaystyle =- \frac{4}{2x\sqrt{x}}[/tex]
Reduce and rewrite:
[tex]\displaystyle =-\frac{2}{x(x^{{}^{1}\! / \! {}_{2} \!})}[/tex]
Simplify:
[tex]\displaystyle =-\frac{2}{x^{{}^{3}\!/\!{}_{2}}}[/tex]
Therefore:
[tex]\displaystyle f'(x)=-\frac{2}{x^{{}^{3}\!/\!{}_{2}}}[/tex]
Related Questions
the thickness of a surfboard relates to the weight of the surfer.a surfboard is 21 3/16 inch thick.write each dimension as a decimal.
Answers
Answer:
https://tex.z-dn.net/?f=%5Cunderline%7B0.1875%7D%5C%5C3%3A16%5C%5C%5Cunderline%7B0%7D%5C%5C30%5C%5C%5Cunderline%7B16%7D%5C%5C140%5C%5C%5Cunderline%7B128%7D%5C%5C.%5C%20120%5C%5C%5Cunderline%7B.%5C%20112%7D%5C%5C.%5C%20%5C%2080%5C%5C%5Cunderline%7B.%5C%20%5C%2080%7D%5C%5CR%3D0Step-by-step explanation:
Highest common factor of 72 and 96
Answers
Answer:
24
Step-by-step explanation:
Prime Factorise 72:
[tex]72=2*36\\72=2*2*18\\72=2*2*2*9\\72=2*2*2*3*3\\\\72=2^3*3^2[/tex]
Prime factorise 96:
[tex]96=2*48\\96=2*2*24\\96=2*2*2*12\\96=2*2*2*2*6\\96=2*2*2*2*2*3\\96=2^5*3[/tex]
Multiply the common factors of [tex]2^3*3^2[/tex] and [tex]2^5*3[/tex]:
[tex]2^3*3=24[/tex]
At midnight, the temperature was 15°C. Over the next six hours, the temperature dropped 3 degrees each hour. By noon, the temperature increased 12 degrees. A. What was the temperature at 6 a.m.? B. Then what was the temperature at noon?
Answers
Answer:
a)3C b)27C
Step-by-step explanation:
a) 6hours*3degrees=18 degrees
15-18=3C
b)15+12=27C
How many seconds are in 4.98 minutes? Round your answer to the nearest tenth (to one decimal point)
Answers
Answer: 298.8 seconds
Step-by-step explanation: 4.98*60=298.8
Answer:
298.8
Step-by-step explanation:
4.98x60=298.8 so you have been understand
Which expression is equivalent to 5(x - 7)?
Answers
Answer:
5x - 35
Step-by-step explanation:
Apply the distributive property. That means to multiply the number on the outside of the perenthesis with each of the numbers inside the perenthesis.
How many signs can be made from a piece of
sheet metal that is 4 feet long and 8 feet wide?
Answers
zelly works 30 hours a week at a food market for $8.50 an hour she takes home $7.65 an hour after deductions. what is her rate for deductions? use an equation to model the problem and solve it. use r to represent the rate for her deductions.
Answers
Answer:
Her deduction is 10%, or 0.85
Step-by-step explanation:
COMPETE
Answer as a dec
2.8+7.22
DONE
Answers
Answer:
Hi, there my name is 1frogaroni and I will be helping you out today!
Step-by-step explanation:
The answer to your question is 10.02.
2.8+7.22
=2.8+7.22
=10.02
You must simplify. if this helped I would be happy to get brainliest in order for me to rank up.
Have a great day!
A bar of soap has a volume of 200ml. It has a mass of 415 grams .what is the density of the bar of soap
Answers
Answer:
8
Step-by-step explanation:
Please Help!! Will mark brainiest!!
Answers
It’s the one that shows 6 is greater than x. So that’s means x is less than 6 not equal to
Whats 855000 in scientific notation?
Answers
Answer:
8.55x10^5 in scientific notation
Answer:
8.55 × 10^5
Step-by-step explanation:
855000 = 855 × 10³
8.55 × 10² × 10³
8.55 × 10²+³
8.55 × 10^5
Adult tickets for a charity concert were $5 each. Students were admitted for $2 each. The charity concert
sold 700 tickets and made $3050. How many students attended the lecture?
Answers
Answer:
students 150
adult 550
Step-by-step explanation:
x + y = 700
5x + 2y = 3050
x = 700 - y
5(700 - y) + 2y = 3050
3500 - 5y + 2y = 3050
-3y + 3500 = 3050
-3y = 3050 - 3500
-3y = -450
y = -450 / -3
y = 150
x = 700 - y
x = 700 - 150
x = 550
Danny wants to sell his house. He would like to receive $136,000 after deducting the 6% commission due to the real estate agent. What should the selling price for the house be to the nearest dollar?
Answers
Answer:
The selling price of the house is $144,681
Step-by-step explanation:
The computation of the selling price of the house is shown below:
Let us assume the selling price be x
So, the commission would be 0.06x
Now the danny would received
= x - 0.06x
= 0.94x
And, the amount received is $136,000
So equate this value
Like this
0.94x = $136,000
x = [tex]\frac{\$134,000}{0.94}[/tex]
= $144,681
Hence, the selling price of the house is $144,681
A dolphin was swimming 28 meters below the ocean's surface. It changed its depth to avoid a predator and ended up 21 meters below the surface.
What was the change in the dolphin's depth?
Answers
Answer:
7 meters
Step-by-step explanation:
Now write an equation that represents the situation. You want to find the change in the dolphin's depth, so subtract its beginning depth from its final depth.
The change in the dolphin's depth is negative 7 meters.
What is Subtraction?Subtraction of two or more numbers or algebraic expressions is the method of taking away a certain value from a number. This operation is also called difference of numbers.
Subtraction is one of the basic operations in mathematics along with division, multiplication and addition.
Initial depth that Dolphin was swimming = 28 meters
Then the Dolphin changed it's depth to avoid predator.
Final depth that Dolphin was swimming = 21 meters
Change in depth = Final depth - Initial depth
= 21 - 28
= -7 meters
This implies that depth of the Dolphin decreases with -7 meters.
Hence the change in the Dolphin's depth is -7 meters if the initial depth was 28 meters and final depth was 21 meters.
To learn more about Subtraction, click on the link given below :
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Which function describes this table of values?
Answers
Answer:
y = 4/5x
Step-by-step explanation:
We know it's not y = x - 2 because if you plug x into the equation, it works for the first one, but not the second one. Ex: 10 - 2 = 8 but 5 - 2 = 3 not 4.
We also know that it can't be y = -5/4x because if you plug x into that equation, it does not equal y.
y = 4/5x because when you plug x into the equation it works.
TLDR basically you just plug x into the equations.
URGENT: how many integers have an absolute value less than 10π
Answers
Answer:
Step-by-step explanation:
Let's calculate 10 pi.
1 pi is about 3.14.
10 pi should be about 31.4.
-31 to -1 is 31 integers.
1 to 31 is 31 more integers.
0 is 1 integer.
31 + 31 = 62
62 + 1 = 63
Hope this helped!
I'm sorry if I misunderstood.
Please mark Brainliest so others can see.
Good luck and try your best everyday :) !
In the given diagram, which of the following can you use to prove that triangle ABC is an isosceles triangle? Show the slopes of AB and CD are negative reciprocals. Show that D is the midpoint of AB. Show that CD = AD = DB.
Answers
Answer:
I’ll tell you that the answer is NOT l, ll, and lll
Step-by-step explanation:
The tempature at 5pm is 20°F. The tempature at 10pm is -5°F. How many degrees did the tempature fall?
Answers
Answer:
25 degrees
Step-by-step explanation:
20 - (-5)
negatives cancel out
20+5 = 25
:D
Which of the following transformations will not produce a congruent image?
A.
rotation
B.
translation
C.
dilation
D.
reflection
Answers
Answer:
Translation, rotation and reflection
Step-by-step explanation:
please brainliest <3
Question 24
The Sugar Sweet Company is going to transport its sugar to market. It will cost $4500 to rent trucks, and it will cost an additional $250 for each ton of sugar
transported.
Let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported. Write an equation relating C to S. Then use this
equation to find the total cost to transport 17 tons of sugar.
Answers
Answer: C=250S+4500
C=250(17) + 4500
C=4250+4500
C=8750
Step-by-step explanation:
As an equation, 250 would be the slope because it is multiplying by the ton and 4500 is the y intercept. If you substitute in 17 for S, then multiply 17 by 250, you get 4250 then add the 4500 and you get 8750 so if S=17, C=8750
The cost of transporting 17 tons of sugar is $8750.
What is an equation?An equation is written in the form of variables and constants separated by the operation of multiplication and division,
An equation states that terms in different forms on both sides of the equality sign are equal.
Multiplication and division do not separate the terms of an equation.
The given information can be related as,
C = 250S + 4500.
Now, when tons of sugar is 17 we'll simply replace S with 17.
∴ C = 250(17) + 4500.
C = 4250 + 4500.
C = 8750.
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Simplify the expression 5-(3x+2)+14x
Answers
Answer:
[tex]=11x+3[/tex]
Step-by-step explanation:
So we have the expression:
[tex]5-(3x+2)+14x[/tex]
Distribute the -1:
[tex]=5-3x-2+14x[/tex]
Combine like terms:
[tex]=(-3x+14x)+(5-2)[/tex]
Add and subtract:
[tex]=11x+3[/tex]
And we're done!
11x + 3
Step-by-step explanation:
5 _(3x + 2) + 14x
5 _ 3x_ 2 + 14x
11x + 3
Multiply the polynomials. (x2 – 4) (x2 + 3x – 5)
Answers
Answer:
[tex] \boxed{ \bold{ \boxed{ \sf{ {x}^{4} + 3 {x}^{3} - 9 {x}^{2} - 12x + 20}}}}[/tex]
Step-by-step explanation:
[tex] \sf{( {x}^{2} - 4)( {x}^{2} + 3x - 5)}[/tex]
Use distributive property
[tex] \dashrightarrow{ \sf{ {x}^{2} ( {x}^{2} + 3x - 5) - 4( {x}^{2} + 3x - 5)}}[/tex]
[tex] \dashrightarrow{ \sf{ {x}^{4} + 3 {x}^{3} - 5 {x}^{2} - 4 {x}^{2} - 12x + 20}}[/tex]
Collect like terms
[tex] \dashrightarrow{ \sf{ {x}^{4} +3 {x}^{3} - 9 {x}^{2} - 12x + 20}}[/tex]
Hope I helped!
Best regards! :D
the quotient of 100 and w
Answers
Answer:
[tex]\frac{100}{w}[/tex]
Step-by-step explanation:
When it asks for the quotient you are dividing/ putting it in a fraction.
Coach Kawa collected data on 5 of his baseball players. The points show shoe size and the number of runs scored this week by 4 of his players.
Answers
Answer:
Paulo and Jason
Step-by-step explanation:
Carmen wears a size 10 so everyone left of Carmen or size 10 (x-axis) are the answers, that would be Paulo and Jason.
Point B is on line segment AC. Given AB = 2x + 10, BC = x, and AC = 5x,
determine the numerical length of AC.
Answers
Answer:
Step-by-step explanation:
the answer is 25 not 5
The numeric value of x will be 25 for the given line segment.
What is a line segment?A line section that can connect two places is referred to as a segment.
In other words, a line segment is just part of a big line that is straight and going unlimited in both directions.
The line is here! It extends endlessly in both directions and has no beginning or conclusion.
Given the line segment AC
AC = AB + BC
Given,
AB = 2x + 10, BC = x, and AC = 5x,
So,
5x = 2x + 10 + x
5x = 3x + 10
2x = 10
x = 5
So,
AC = 5x
AC = 5(5) = 25 units.
Hence, The numeric value of x will be 25 for the given line segment.
For more about line segment
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Which of the following numbers can be classified as a whole number, natural number, integer, and rational number? 0.4. -78 15 0
Answers
Answer:
0.4 since the number is non terminating and non repeating therefore it is irrational number.
-78 A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero
15 The number 15 is a rational number. It is an integer and all integers are rational numbers.
0 Yes zero is a rational number. Thus, 0 can be written as, where a/b = 0, where a = 0 and b is any non-zero integer.
Jillian said √25 is irrational because it is a square root. Why is she wrong
Answers
Answer:
She is wrong because the answer is 5 and that is rational
Step-by-step explanation:
The quotient of 24 and x equals 14 minus 2 times x
Answers
Answer:
x = 3, x = 4
Step-by-step explanation:
Given
[tex]\frac{24}{x}[/tex] = 14 - 2x ( multiply through by x , x ≠ 0 )
24 = 14x - 2x² ( subtract 14x - 2x² from both sides )
2x² - 14x + 24 = 0 ( divide through by 2 )
x² - 7x + 12 = 0 ← in standard form
(x - 3)(x - 4) = 0 ← in factored form
Equate each factor to zero and solve for x
x - 3 = 0 ⇒ x = 3
x - 4 = 0 ⇒ x = 4
I need to answer this using dimensional analysis and the answer must be written in Scientific notation. I can't figure it out.
Answers
Answer:
3.343481472 x 10^4
Step-by-step explanation:
11.6 a = 3.25 *
6.18 * = 2.5 #
Take the first equation and solve for *
11.6 a = 3.25 *
Divide each side by 3.25
11.6 a/3.25 = *
3.569230769 a = *
Substitute this into the second equation for *
6.18 (3.569230769 a) = 2.5 #
22.05784615 a = 2.5 #
Solve for a
Divide by 22.05784615
a = 2.5 # / 22.05784615
a = .113338355
We have 2.95 x 10 ^5 a
2.95 x 10 ^5 ( .113338355)
33434.81472
Changing to scientific notation
3.343481472 x 10^4
Solve the following equation 2/3x+6=1/2x+1/4x
Answers
Answer:
x = -72/17
Step-by-step explanation:
(2/3)*x+(1/2)*x+(1/4)*x+6 = 0
17/12*x+6 = 0 // - 6
17/12*x = -6 // : 17/12
x = -6/17/12
x = -72/17
x = -72/17