Use the distributive property to find the product

Answer:

[tex]\displaystyle \frac{dy}{dx}\Big|_{(1, 1)}=0[/tex]

Step-by-step explanation:

We are given the equation:

[tex](x^2+y^2)^2=4x^2y[/tex]

And we want to find the slope of the tangent line at the point (1,1).

Recall that the slope of the tangent line to a point of a function is given by the function's derivative.  

Thus, find the derivative of the equation. Take the derivative of both sides with respect to x:

[tex]\displaystyle \frac{d}{dx}\left[(x^2+y^2)^2\right]=\frac{d}{dx}\left[4x^2y\right][/tex]

Let's do each side individually.

Left:

We can use the chain rule:

[tex](u(v(x))'=u'(v(x))\cdot v'(x)[/tex]

Let v(x) be x² + y² and u(x) is x². Thus, u'(x) is 2x. Therefore:

[tex]\displaystyle \frac{d}{dx}\left[(x^2+y^2)^2\right]=2\left(x^2+y^2\right)\left(\frac{d}{dx}\left[x^2+y^2\right]\right)[/tex]

Differentiate:

[tex]\displaystyle \frac{d}{dx}[(x^2+y^2)^2]=2(x^2+y^2)\left(2x+2y\frac{dy}{dx}\right)[/tex]

Factor. Therefore, our left side is:

[tex]\displaystyle 4(x^2+y^2)\left(x+y\frac{dy}{dx}\right)[/tex]

Right:

We have:

[tex]\displaystyle \frac{d}{dx}\left[4x^2y\right][/tex]

We can move the constant multiple outside:

[tex]\displaystyle =4\frac{d}{dx}\left[x^2y\right][/tex]

We will use the Product Rule:

[tex]\displaystyle =4\left(\frac{d}{dx}\left[x^2\right]y+x^2\frac{d}{dx}\left[y\right]\right)[/tex]

Differentiate:

[tex]\displaystyle =4\left(2xy+x^2\frac{dy}{dx}\right)[/tex]

Therefore, our entire equation is:

[tex]\displaystyle 4(x^2+y^2)\left(x+y\frac{dy}{dx}\right)=4\left(2xy+x^2\frac{dy}{dx}\right)[/tex]

To find the derivative at (1,1), substitute and evaluate dy/dx when x = 1 and y = 1. Hence:

[tex]\displaystyle 4((1)^2+(1)^2)\left((1)+(1)\frac{dy}{dx}\right)=4\left(2(1)(1)+(1)^2\frac{dy}{dx}\right)[/tex]

Evaluate:

[tex]\displaystyle 4((1)+(1))\left(1+\frac{dy}{dx}\right)=4\left(2+\frac{dy}{dx}\right)[/tex]

Further simplify:

[tex]\displaystyle 8\left(1+\frac{dy}{dx}\right)=8+4\frac{dy}{dx}[/tex]

Distribute:

[tex]\displaystyle 8+8\frac{dy}{dx}=8+4\frac{dy}{dx}[/tex]

Solve for dy/dx:

[tex]\displaystyle 4\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=0[/tex]

Therefore, the slope of the tangent line at the point (1, 1) is 0.

There are two terms x^2 is one and 121 is the other one.

The answer is 2 terms.

Answer: $1,104

Step-by-step explanation:At the end of 20 years, your savings will have grown to $1,604.

You will have earned in $1,104 in interest.

The answer is -14. 0+8 =8 8 - (6x4) = - (-16 + 2 = -14

Answer:

-14

Step-by-step explanation:

Answer:

a

Step-by-step explanation:

The equation which can be represented using the picture displayed on the number line is Three-fourths divided by one-eighth = 6

The number of steps or marked points can be obtained using the relation :

Length of line ÷ Distance between each marked points

3/4 ÷ 1/8 = 6

Therefore, the expression which represents the scenario on the number line is 3/4 ÷ 1/8 = 6

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Answer:

m ≥ 3

m is greather or equal to 3

Step-by-step explanation:

m + 4 ≥ 7

m ≥ 7 - 4

m ≥ 3

Answer:

the missing value is 9.33.

Step-by-step explanation:

given,

length (l) = 3x+2

breadth (b) = 4

perimeter(P) = 68 cm

now,

P = 2(l+b)

or, 68 = 2× ( 3x+2+4)

or, 68 = 2× ( 3x+ 6)

or, 68 = 6x+12

or, 68 - 12= 6x

or, 56 = 6x

or, 6x = 56

or, x = 56/6

or, x = 9.33

Answer:

b = $48000, m = $3187.5 / year

Step-by-step explanation:

The equation of a linear function is given as y = mx + b, where m is the rate of change, b is the value of y when x = 0, y = dependent variable and x = dependent variable.

Given that V = mt + b:

b = initial price of the house at 0 years = $48000

V = mt + 48000, At 8 years the house is appraised at ​$73,500

73500 = 8m + 48000

8m = 73500 - 48000

8m = 25500

m = 3187.5

Answer:

6+7x

Step-by-step explanation:

Answer:

26

Step-by-step explanation:

1/6 of 36 is 6

1/9 of 36 is 4

6+4=10

36-10= 26

Answer:

26 cakes

Step-by-step explanation:

36 divided by 6 = 6

30 - 6 = 24

36 divided by 9 = 4

6 + 4 = 10

36 - 10 = 26

Hopefully this helps you :)

pls give me brainlest  ;)

Answer:  Any equation that is of the form y = -3/4x + b, where b is a real number, will be parallel to y = -3/4x - 2.

Step-by-step explanation:

A parallel line has the same slope as the original line.  Therefore, if the equation of the original line is y = mx+b, m (the slope) will remain the same, while b can change.

The equation  y = -3/4x - 2 has the same slope -3/4, represents a line which is parallel to the line y=-3/4 x -2

A function that can be represented in the form of y =mx +c, is called a Linear Function. where m is the slope and c is the y-intercept.

The slope of the given line can be found as;

A parallel line has the same slope as the original line.  Thus, if the equation of the original line is y = mx+b, m (the slope) will remain the same, while b can change.

Any equation that is of the form y = -3/4x + b,

where b is a real number, will be parallel to the equation.

y = -3/4x - 2.

m = -3/4

The equation  y = -3/4x - 2 has the same slope -3/4, represents a line which is parallel to the line y=-3/4 x -2

To know more about Linear Functions;

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Answer:

5 in display, 7 in each aquarium.

Step-by-step explanation:

Answer:

b = -16

Step-by-step explanation:

9b = 8b - 16

9b - 8b = -16

b = -16

Answer:

9

Step-by-step explanation:

If three magazines cost $15 then that means that each magazine costs $5. So we do 5 divided by 45 which equals 9. SO the answer is 9!

If you are happy with my answer I would like branliest:)

Answer:

2 builder complete work = 155 days

Step-by-step explanation:

Given:

5 builder complete work = 62 days

So,

1 builder complete work = 62 days × 5

if,

2 builder complete work = [62 days × 5] / 2

2 builder complete work = 155 days

Answer:

d is correct

Step-by-step explanation:

Answer:

This is a quadratic function, which is usually defined on all reals. But, the domain is restricted to x>=0 because it does not make sense to produce a negative number of units.

Find the vertex of the parabola at

(10.14, 164.5)

Since the parabola opens up, the range is y>=164.5

Step-by-step explanation:

Answer: -3

Step-by-step explanation:

Since we are given the value of r, we can plug that value into the expression and solve.

15+6(-3)              [multiply]

15-18                   [subtract]

-3

Now, we know that the expresison is equal to -3.

Answer:

A.)  What natural environments are in an area

Step-by-step explanation:

Answer:

k < 8

Step-by-step explanation:

-7k + 30k + 25 < 209

23k < 209 - 25

23k < 184 /: 23

k < 8

If you have any questions about the way I solved it, don't hesitate to ask me in the comments below ÷)

Answer:

$250 per month  for 2 years

Step-by-step explanation:

The answer is x=2

move all terms that dont contain x to the right side and then solve.

Answer:

x=2

Step-by-step explanation:

Sorry the picture is sideways, I was trying to fit everything in lol.

Answer:

Step-by-step explanation: yes

Answer:

8.3

Step-by-step explanation:

hope it helps

where is the number line

Step-by-step explanation:

Answer:

8+ 1/2 (12) = x

8+ 6 = x

Therefore x = 14

Answer:

The answer is X = 14

Step-by-step explanation:

Step 1: Simplify both sides of the equation.

Step 2: Flip the equation.

Edgenuity’s sample response for this question:

Sublimation occurs because the atoms in a solid must gain enough energy to form a gass without passing through the liquid state. Low air pressure and large amounts of energy, often in the form of heat, allow the atoms to move fast enough to overcome the attractive forces between them.

If you need an original response here’s my take!!:

Sublimation happens when a solid turns directly into a gas. For this to happen, there must be low air pressure and immense heat energy. This allows atoms to quickly move and perform sublimation.

[tex]3x^2-10xy + 8y^2=\\\\=3x^2-6xy-4xy + 8y^2=\\\\=3x(x-2y)-4y(x-2y)=\\\\=(x-2y)(3x-4y)[/tex]

Answer:  

For part A its called a ray.For part B it would be EA.

Step-by-step explanation:

Part A would be called a ray because B is the endpoint while C is continuous and a ray has only one endpoint.Part B is EA because E would be the endpoint while A would be continuous