find from first principle the derivative of 3x+5/√x​

Answers

Answer 1

Answer:

[tex]\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}[/tex]

General Formulas and Concepts:

Algebra I

Exponential Rule [Powering]:                                                                          [tex]\displaystyle (b^m)^n = b^{m \cdot n}[/tex]Exponential Rule [Rewrite]:                                                                              [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex] Exponential Rule [Root Rewrite]:                                                                     [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]

Calculus

Derivatives

Derivative Notation

Derivative Property [Addition/Subtraction]:                                                            [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                               [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \frac{3x + 5}{\sqrt{x}}[/tex]

Step 2: Differentiate

Rewrite [Exponential Rule - Root Rewrite]:                                                     [tex]\displaystyle \frac{3x + 5}{x^\bigg{\frac{1}{2}}}[/tex]Quotient Rule:                                                                                                   [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{(x^\bigg{\frac{1}{2}})^2}[/tex]Simplify [Exponential Rule - Powering]:                                                          [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{x}[/tex]Basic Power Rule [Derivative Property - Addition/Subtraction]:                   [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})(3x^{1 - 1} + 0) - (\frac{1}{2}x^\bigg{\frac{1}{2} - 1})(3x + 5)}{x}[/tex]Simplify:                                                                                                             [tex]\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2}x^\bigg{\frac{-1}{2}})(3x + 5)}{x}[/tex]Rewrite [Exponential Rule - Rewrite]:                                                              [tex]\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2x^{\frac{1}{2}}})(3x + 5)}{x}[/tex]Rewrite [Exponential Rule - Root Rewrite]:                                                     [tex]\displaystyle \frac{d}{dx} = \frac{3\sqrt{x} - (\frac{1}{2\sqrt{x}})(3x + 5)}{x}[/tex]Simplify [Rationalize]:                                                                                       [tex]\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e


Related Questions

T
On Melissa's 6th birthday, she gets a $2000 CD that earns 5% interest, compounded semiannually. If the
CD matures on her 16th birthday, how much money will be available?
TE
$
(S

Answers

9514 1404 393

Answer:

  $3277.23

Step-by-step explanation:

The future value of the CD with interest at rate r compounded semiannually for t years will be given by ...

  A = P(1 +r/2)^(2t)

where P is the principal value.

For the given rate and time, this is ...

  A = $2000(1 +0.05/2)^(2·10) = $2000(1.025^20) ≈ $3277.23

The value of the CD at maturity will be $3277.23.

12.5% of it is 6 hrs​

Answers

Answer:

48 hours

Step-by-step explanation:

12.5/100 = 6/x

1. cross-multiply

12.5 × x = 12.5x

100 × 6 = 600

2. divide

12.5x/12.5 = x

600/12.5 = 48

3. answer

x = 48

How do we solve this? Please help?

Answers

Answer:

f'(1) = 7/2

Step-by-step explanation:

We are given the function: [tex]f'(x)=\frac{7}{2\sqrt{x} }[/tex]

To find f'(1), substitute the value for x and evaluate it.

[tex]f'(1)=\frac{7}{2\sqrt{1} }\\\\f'(1)=\frac{7}{2(1)}\\\\f'(1)=\frac{7}{2}[/tex]

Therefore, f'(1) = 7/2.

Which function represents the graph of f(x)=−4|x| after it is translated 3 units down?

Answers

Answer:

The function with blue line (see attachment) represents the graph of [tex]f(x) = -4\cdot |x|[/tex] after it is translated 3 units down.

Step-by-step explanation:

Mathematically speaking, a translation in the y-direction is defined by the following expression:

[tex]g(x) = f(x) + k[/tex] (1)

Where:

[tex]g(x)[/tex] - Resulting function.

[tex]f(x)[/tex] - Original function.

[tex]k[/tex] - Translation factor ([tex]k > 0[/tex] - Translation in the +y direction/[tex]k< 0[/tex] - Translation in the -y direction).

If we know that original function is translated 3 units down and [tex]f(x) = -4\cdot |x|[/tex], then the resulting function is:

[tex]g(x) = -4\cdot |x|-3[/tex]

Lastly, we graph both original and resulting functions with the help of a graphing tool, whose outcome is presented in the image attached below:

Please notice that the red line represents the original function, whereas the blue line represents the resulting function.

10
8
12
10
14
?

a. 16
b. 10
c. 12
d. 18

Answers

Answer:

12

Step-by-step explanation:

10  8  subtract 2

8 12  add 4

12-10  subtract 2

10 14  add 4

Now we subtract 2

14-2 = 12

a store sign reads "Take 75% of the original price when you take an additional 15% off the sale price, which is 60% off the original price." Is the stores sign accurate?

Answers

Answer:

The new price is 66% off the original not 75% off

Step-by-step explanation:

Let x be the original price

First take 60 percent off

x - x*60% = new price

x- .60x = .40x

The new price is .40x

Then take 15 % off

(.40x) - (.40x)*15%

.40x - .40x*.15

.40x - .06x

.34x

100 -.34 =.66

The new price is 66% off the original not 75% off

For what numbers is f(0) = sec 0 not defined? ​

Answers

Your mom your dad and you brother

Answer:

stundeez

Step-by-step explanation:

Nicki Minaj hdhsbskdhsnsk

If 2x - 5y – 7 = 0 is perpendicular to the line ax - y - 3 = 0 what is the value of a ?

A) a =2/3

B) a =5/2

C) a = -2/3

D) a = -5/2

Answers

Answer:

D) a = - 5/2

Step-by-step explanation:

2x -5y - 7 = 0

5y = 2x - 7

y = 2/5 x - 7

the slope of this line is therefore 2/5 (factor of x).

the perpendicular slope is then (exchange y and x and flip the sign) -5/2, which is then a and the factor of x.

11. Find the equation of the straight line that passes through (-4,2), (3,2) and (7,2).​

Answers

Step-by-step explanation:

[tex](2 - 1 \frac{3 {3}^{2} }{?} [/tex]

Answer:

[tex]y=2[/tex]

Step-by-step explanation:

To solve this question, we can use the point-slope formula as we are given points on the line. This is the format:

[tex]y-y_{1} =m(x-x_{1} )[/tex]

The first step is to find the slope by substituting in two of the points. Let's try using (7,2) and (3,2):

[tex]2-2=m(7-3)\\0=4m\\m=0[/tex]

So now we have found that our slope is 0 meaning it is a flat line (shown by the unchanging y values through all three points).

The form for line equations is:

[tex]y=mx+c[/tex]

However, since our m=0, it is simplified to this:

[tex]y=c[/tex]

The y-value for all the points is 2, meaning c=2 (as it is also the y-intercept)

Therefore our final equation is:

[tex]y=2[/tex]

Hope this helped!

Complete the table y+1=7/8x

Answers

Answer:

y = 7/8x - 1

Step-by-step explanation:

graph shown

Find the area If you get this correct i WILL GIVE YOU 100 POINTS

Answers

Answer:

Area of yellow portion =54 in

Point D is 8 units away from the origin along the x-axis, and is 6 units away along the y-axis. Which of the following could be the coordinates of Point D

Answers

Answer:

The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

Now we know that point D, which we can write as (x, y), is at a distance of 8 units from the origin.

Where the origin is written as (0, 0)

We also know that point D is 6 units away along the y-axis.

Then point D could be:

(x, 6)

or

(x, -6)

Now, let's find the x-value for each case, we need to solve:

[tex]8 = \sqrt{(x - 0)^2 + (\pm6 - 0)^2}[/tex]

notice that because we have an even power, we will get the same value of x, regardless of which y value we choose.

[tex]8 = \sqrt{x^2 + 36} \\\\8^2 = x^2 + 36\\64 - 36 = x^2\\28 = x^2\\\pm\sqrt{28} = x\\\pm 5.29 = x[/tex]

So we have two possible values of x.

x = 5.29

and

x = -5.29

Then the points that are at a distance of 8 units from the origin, and that are 6 units away along the y-axis are:

(5.29, 6)

(5.29, -6)

(-5.29, 6)

(-5.29, -6)

(a) Find a vector parallel to the line of intersection of the planes −4x+2y−z=1 and 3x−2y+2z=1.
v⃗ =
(b) Show that the point (−1,−1,1) lies on both planes. Then find a vector parametric equation for the line of intersection.
r⃗ (t)=

Answers

Find the intersection of the two planes. Do this by solving for z in terms of x and y ; then solve for y in terms of x ; then again for z but only in terms of x.

-4x + 2y - z = 1   ==>   z = -4x + 2y - 1

3x - 2y + 2z = 1   ==>   z = (1 - 3x + 2y)/2

==>   -4x + 2y - 1 = (1 - 3x + 2y)/2

==>   -8x + 4y - 2 = 1 - 3x + 2y

==>   -5x + 2y = 3

==>   y = (3 + 5x)/2

==>   z = -4x + 2 (3 + 5x)/2 - 1 = x + 2

So if we take x = t, the line of intersection is parameterized by

r(t) = ⟨t, (3 + 5t )/2, 2 + t

Just to not have to work with fractions, scale this by a factor of 2, so that

r(t) = ⟨2t, 3 + 5t, 4 + 2t

(a) The tangent vector to r(t) is parallel to this line, so you can use

v = dr/dt = d/dt ⟨2t, 3 + 5t, 4 + 2t⟩ = ⟨2, 5, 2⟩

or any scalar multiple of this.

(b) (-1, -1, 1) indeed lies in both planes. Plug in x = -1, y = 1, and z = 1 to both plane equations to see this for yourself. We already found the parameterization for the intersection,

r(t) = ⟨2t, 3 + 5t, 4 + 2t

*WILL GIVE BRAINLIEST* Find the length of side xx in simplest radical form with a rational denominator.

Answers

Answer:

[tex]x = \frac{4 \sqrt{3} }{3} [/tex]

Step-by-step explanation:

The explanation is in the picture!❤

what is the prime product of 120

Answers

Answer:

[tex]2^{3} * 3 * 5[/tex]

Step-by-step explanation:

ABC are points; (2,3), (4,7), (7,3) respectively. Find the equation of the line through the point (3,-5) which is parallel to the line with the equation 3x+2y-5=0​

Answers

Answer:

y = -3x/2 - 1/2

Step-by-step explanation:

slope m = -3/2

-5 = (-3/2)×3+b

or, b = -1/2

putting it into y = mx + b

y = -3x/2 - 1/2

Answered by GAUTHMATH

Complete the equation of this circle:
pleaseee helpp

Answers

Answer:

(x – 3)^2+ (y – 2)^2 = r^2

Step-by-step explanation:

The co-ordinates of the center=(3,2)

The formula for the equation of a circle is (x – h)^2+ (y – k)^2 = r^2,

(h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle.

Therefore, (x – h)^2+ (y – k)^2 = r^2

                 =(x – 3)^2+ (y – 2)^2 = r^2

You can find r^2the equation.

The length side of xy is?

Answers

Answer:

10

Step-by-step explanation:

ok so you do 12/30 and u get a 0.4 ratio. boom multiply 0.4 by 25 and u get 10. so boom the length is 10

Answer:

XY=10

Step-by-step explanation:

Since they are similar the ratio between each sides should be the same.

Ratio is .4. Found by dividing 12/30.

Multiply .4 by 25= 10

What is 3.142857 rounded three decimal places

Answers

Answer:

3.143

Step-by-step explanation:

Since you're rounded it to the third decimal, you look at the fourth one. And since the fourth one is 8, ur going round 2, which is the third decimal to 3.

Answer:

The answer is below

Step-by-step explanation:

In this question, to round three decimal place, we need to count three numbers after the dot.

Here..

the three numbers are 142

so after that we round it, so first of all we ignore 1 and, and focus on the number 2 and 4.

and if the number is below 5, it is rounded to the last number, but if the number is 5 or more than it, then we round it to the next number.

For example

142 - here we have 2, which is below 5, so we round it to '140'

where as if we have..

147 - here we have 7, which is above 7, so we round it to '150'

____________________________________________________________

So in this problem we round 142, so the number 2 is below 5 so it is round to 140

therefore the answer is - 3.14

Pls help I literally am crying I don’t understand ):

Answers

Answer: Passes out in slow

Step-by-step explanation:

Step 1 be Einstein

Ayuda por fa con estos ejercicios por fa urgente

Answers

Step-by-step explanation:

A ball is thrown straight up from a rooftop 320 feet high. The formula below describes the ball's height above the ground, h, in feet, t seconds after it was thrown. The ball misses the rooftop on its way down and eventually strikes the ground. How long will it take for the ball to hit the ground? Use this information to provide tick marks with appropriate numbers along the horizontal axis in the figure shown.

h=-16t^2+16t+32

In an experiment, you choose to have two randomly assigned groups. In one, you take measurements both pretest and posttest; with the second, a posttest-only measure. This describes which task of conducting an experiment

Answers

Answer:

The answer is "Specific treatment levels".

Step-by-step explanation:

When we experimenting with 'level' which is related to the quantity or magnitude of treatment. For this part of an experiment or study, a group or individual is exposed to a specified set of circumstances. For example: If four categories are exposed to different doses of a given drug, then each dose reflects a level of a treatment factor in the model.

CAN SOMEONE HELP ME PLEASE CAN YOU FIGURE OUT WHERE I PUT 4 PI ON THE NUMBER LINE

Answers

Answer:

see below

Step-by-step explanation:

Pi is approximately 3.14

4*3.14 =12.56

So about halfway between 12 and 13

A hose is left running for 240 minutes to 2 significant figures. The amount of water coming out of the hose each minute is 2.1 litres to 2 significant figures. Calculate the lower and upper bounds of the total amount of water that comes out of the hose.​

Answers

Answer:

Hello,

Step-by-step explanation:

Let say t the time the hose is left running

235 ≤ t < 245 (in min)

Let say d the amount of water coming out of the hose each minute

2.05 ≤ d < 2.15 (why d : débit in french)

235*2.05 ≤ t*d < 245*2.15

481.75 ≤ t*d < 526.75    (litres)

Answer:

Lower bound: [tex]495\; \rm L[/tex] (inclusive.)

Upper bound: [tex]505\; \rm L[/tex] (exclusive.)

Step-by-step explanation:

The amount of water from the hose is the product of time and the rate at which water comes out.

When multiplying two numbers, the product would have as many significant figures as the less accurate factor.

In this example, both factors are accurate to two significant figures. Hence, the product would also be accurate to two significant figures. That is:

[tex]240 \times 2.1 = 5.0 \times 10^{2}\; \rm L[/tex] ([tex]500\; \rm L[/tex] with only two significant figures.)

Let [tex]x[/tex] denote the amount of water in liters. For [tex]x\![/tex] to round to [tex]5.0 \times 10^{2}\; \rm L[/tex] only two significant figures are kept, [tex]495 \le x < 505[/tex]. That gives a bound on the quantity of water from the hose.

surface area of a prism please help its my last day 120 points

Answers

Answer:

Area of the base = (8×6)/2 = 24 yd²

Height of the prism = 8 yd

Perimeter of the base = 8+6+10 = 24 yd

Surface area = 2B + Ph = (2×24)+(24×8) = 48+192 = 240 yd²

On the first day of a two-day meeting, 10 coffees and 10 doughnuts were purchased for a total of $20.00. Since nobody drank the coffee and all the doughnuts were eaten, the next day only 2 coffees and 14 doughnuts were purchased for a total of $13.00. How much did each coffee and each doughnut cost?

Answers

Answer:

1.25 dollars- the value of each coffee

0.75 dollars- the value of each doughnut

Step-by-step explanation:

Suppose that value of one coffee is x dollars, when one doughnut costs y dollars. The value o 10 coffee is 10x, when 10 doughnuts cost 10y. The sum is 10x+10y and it is 20.

10x+10y= 20 (we can divide each part by 10)

x+y=2

2coffee cost  2x, 14 doughnuts cost 14y

2x+14y=13 (it can be divided by two)

x+7y=6.5

We have the system of equations x+y=2,  x+7y=6.5

Subtract the first equation from the second one (the left side of the first equation from the left side

x+7y - (x+y)= 6.5-2

6y=4.5

y=4.5/6= 3/4 = 0.75 dollars- the value of each doughnut

x=2-0.75=1.25 dollars- the value of each coffee

Algebra 2, please help! thank you
The function y = 2 cos 3(x + 2π∕3) +1 has a phase shift (or horizontal shift) of

A) –2π∕3
B) 3
C) 1
D) 2

Answers

Answer:

-2pi/3

Step-by-step explanation:

y = 2 cos 3(x + 2π∕3) +1

y = A sin(B(x + C)) + D  

amplitude is A

period is 2π/B

phase shift is C (positive is to the left)

vertical shift is D

We have a shift to the left of 2 pi /3

Answer:

A

Step-by-step explanation:

The standard cosine function has the form:

[tex]\displaystyle y = a\cos (b(x-c)) + d[/tex]

Where |a| is the amplitude, 2π / b is the period, c is the phase shift, and d is the vertical shift.

We have the function:

[tex]\displaystyle y = 2 \cos 3\left(x + \frac{2\pi}{3}\right) + 1[/tex]

We can rewrite this as:

[tex]\displaystyle y = \left(2\right)\cos 3\left(x - \left(-\frac{2\pi}{3}\right)\right) + 1[/tex]

Therefore, a = 2, b = 3, c = -2π/3, and d = 1.

Our phase shift is represented by c. Thus, the phase shift is -2π/3.

Our answer is A.

1. Choose the correct decimal for "three tenths."
3
0.03
0.003
0.3
Please hurry, if you do reply thank u, it means alot! <3 :)

Answers

Answer:

3 tenths means 3 over ten represented as as 3/10 and 10 has one zero I.e tenth different from hundredths which has 2 zeros so our decimal shld also have one zero which is 0.3...so 0.3 is the answe hope it helps❤

Help? Thanks!!!!!!!! If possible show work please!

Answers

Answer:

[tex]B =102[/tex]

[tex]Y = 32[/tex]

Step-by-step explanation:

Solving (47):

To solve for B, we have:

[tex]B + 50 + 28 = 180[/tex] --- sum of angles in a triangle

This gives

[tex]B + 78 = 180[/tex]

Collect like terms

[tex]B =- 78 + 180[/tex]

[tex]B =102[/tex]

Solving (48):

To solve for Y, we have:

[tex]X + Y+ Z = 180[/tex] --- sum of angles in a triangle

This gives

[tex]Y = 180 - X - Z[/tex]

Where

[tex]W+ X=180[/tex] -- angle on a straight line

Solve for X

[tex]X=180 -W[/tex]

[tex]X=180 -100 = 80[/tex]

So, we have:

[tex]Y = 180 - X - Z[/tex]

[tex]Y = 180 - 80 - 68[/tex]

[tex]Y = 32[/tex]

help me please i’ll give brainliest the

Answers

Answer:

y=-1/2x+-1

Step-by-step explanation:

try desmos with this equation.

y=mx+b

m=the slope which is -1/2. It goes down 1 it is negative because it is going down, and to the right 2.

b=y-intercept meaning the point which the line crosses the line y .-1

Other Questions
PLEASE help me ASAP. Social studies question.1. How can we make proper use of electricity? present any six ways. What is the surface area of this figure in square centimeters?A.96B.75C.84D.60 Please help me with this question, thank you Phn tch nguyn tc xc nh thm quyn x pht vi phm hnh chnh quy nh ti iu 52 Lut x l vi phm hnh chnh nm 2012. Ly v d cho tng khon Which equation is the inverse of y = 16+ 1?Choices made this question if you can able to make Factor the polynomial expression 3x4 + 24x. Find the length of the third side. If necessary, round to the nearest tenth.1520Submit AnswerAnswer:attempt 1 out of 2PLS HELP ASAP ASAP ITS TIMED What is the following sum? Assume x20 and 20.W x + 2 / x 374 + xy y0 x 7 y- 2xy?0 2xy y + 2xy x0 4xy dxo 2xyxy The endpoints (horizontal and vertical intercepts) of the budget line: Group of answer choices represent the quantity of each good that could be purchased if all of the budget were allocated to that good. measure the rate at which one good can be substituted for another. measure its slope. indicate the highest level of satisfaction the consumer can achieve. measure the rate at which a consumer is willing to trade one good for another. in a 5 liter milk jar water and milk are in the ratio 2:3. If 1 liter of milk and 1 liter of water is added. what will be the new ratio.Give me answer with steps . and the answer is 3:4. but how I want to know Who does the Time Traveller meet when he lands in the future in The Time Machine?A. People dressed in purpleB. Wild lifecougars and lionsC. A group of masked figuresD. Green aliens the more beautiful or handsome you are the more problem you have.argue for or against this statement What group of people might the Medical Man represent in The Time Machine?A. DoctorsB. LawyersC. PoliticiansD. Teenagers A longshore current generally runs __________. a. parallel to the coastline b. perpendicular to the coastline c. against the prevailing wind direction. d. parallel to the prevailing wind direction. A sample from a stock of a bacterial colony in liquid media was diluted by a factor of 106, and 2 ml of this dilution was spread on a Petri dish of solidified media. 56 colonies were observed. What was the concentration of bacteria of the initial stock? Is the United States a civilization? mention 6 substances secreted by the body State examples of a transverse wave. Assuming x and y are two odd numbers and x/y is an integer. Which of the following statements are true?1: x + y is odd. 2: xy is odd. 3: x/y is odd. 4: x-y is odd. A: 1 and 2 only. B: 1 and 3 only. C: 2 and 3 only. D: 2, 3, and 4 only. E: 2 and 4 only.