How can you multiply fractions and mixed numbers?​

(i) I would first suggest writing this function as a product of the functions,

[tex]\displaystyle y = fgh = (4+3x^2)^{1/2} (x^2+1)^{-1/3} \pi^x[/tex]

then apply the product rule. Hopefully it's clear which function each of f, g, and h refer to.

We then have, using the power and chain rules,

[tex]\displaystyle \frac{df}{dx} = \frac12 (4+3x^2)^{-1/2} \cdot 6x = \frac{3x}{(4+3x^2)^{1/2}}[/tex]

[tex]\displaystyle \frac{dg}{dx} = -\frac13 (x^2+1)^{-4/3} \cdot 2x = -\frac{2x}{3(x^2+1)^{4/3}}[/tex]

For the third function, we first rewrite in terms of the logarithmic and the exponential functions,

[tex]h = \pi^x = e^{\ln(\pi^x)} = e^{\ln(\pi)x}[/tex]

Then by the chain rule,

[tex]\displaystyle \frac{dh}{dx} = e^{\ln(\pi)x} \cdot \ln(\pi) = \ln(\pi) \pi^x[/tex]

By the product rule, we have

[tex]\displaystyle \frac{dy}{dx} = \frac{df}{dx}gh + f\frac{dg}{dx}h + fg\frac{dh}{dx}[/tex]

[tex]\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} (x^2+1)^{-1/3} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} (x^2+1)^{-1/3} \ln(\pi) \pi^x[/tex]

[tex]\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} \frac{1}{(x^2+1)^{1/3}} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} \frac{1}{ (x^2+1)^{1/3}} \ln(\pi) \pi^x[/tex]

[tex]\displaystyle \frac{dy}{dx} = \boxed{\frac{\pi^x}{(4+3x^2)^{1/2} (x^2+1)^{1/3}} \left( 3x - \frac{2x(4+3x^2)}{3(x^2+1)} + (4+3x^2)\ln(\pi)\right)}[/tex]

You could simplify this further if you like.

In Mathematica, you can confirm this by running

D[(4+3x^2)^(1/2) (x^2+1)^(-1/3) Pi^x, x]

The immediate result likely won't match up with what we found earlier, so you could try getting a result that more closely resembles it by following up with Simplify or FullSimplify, as in

FullSimplify[%]

(% refers to the last output)

If it still doesn't match, you can try running

Reduce[<our result> == %, {}]

and if our answer is indeed correct, this will return True. (I don't have access to M at the moment, so I can't check for myself.)

(ii) Given

[tex]\displaystyle \frac{xy^3}{1+\sec(y)} = e^{xy}[/tex]

differentiating both sides with respect to x by the quotient and chain rules, taking y = y(x), gives

[tex]\displaystyle \frac{(1+\sec(y))\left(y^3+3xy^2 \frac{dy}{dx}\right) - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = e^{xy} \left(y + x\frac{dy}{dx}\right)[/tex]

[tex]\displaystyle \frac{y^3(1+\sec(y)) + 3xy^2(1+\sec(y)) \frac{dy}{dx} - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = ye^{xy} + xe^{xy}\frac{dy}{dx}[/tex]

[tex]\displaystyle \frac{y^3}{1+\sec(y)} + \frac{3xy^2}{1+\sec(y)} \frac{dy}{dx} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} \frac{dy}{dx} = ye^{xy} + xe^{xy}\frac{dy}{dx}[/tex]

[tex]\displaystyle \left(\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}\right) \frac{dy}{dx}= ye^{xy} - \frac{y^3}{1+\sec(y)}[/tex]

[tex]\displaystyle \frac{dy}{dx}= \frac{ye^{xy} - \frac{y^3}{1+\sec(y)}}{\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}}[/tex]

which could be simplified further if you wish.

In M, off the top of my head I would suggest verifying this solution by

Solve[D[x*y[x]^3/(1 + Sec[y[x]]) == E^(x*y[x]), x], y'[x]]

but I'm not entirely sure that will work. If you're using version 12 or older (you can check by running $Version), you can use a ResourceFunction,

ResourceFunction["ImplicitD"][<our equation>, x]

but I'm not completely confident that I have the right syntax, so you might want to consult the documentation.

Answer:

N=7.2

Step-by-step explanation:

6x-10= -60

10x7.2= 72

72+-60=12

So N=7.2

Answer:

-x² + 4x - 8

Step-by-step explanation:

Given: 2x² - x + 6 is subtracted from x² + 3x - 2

Rewriting: (x² + 3x - 2) - (2x² - x + 6)

Distrubte negative: x² + 3x - 2 - 2x² + x - 6

Combine like terms:  -x² + 4x - 8

Answer: -x² + 4x - 8

Hope this helps, have a nice day :D

Answer:

A

Step-by-step explanation:

Each colored box cost 0.01

Answer:

A

Step-by-step explanation:

Each colored box cost 0.01

let's firstly convert the mixed fractions to improper fractions and then divide.

[tex]\stackrel{mixed}{1\frac{1}{3}}\implies \cfrac{1\cdot 3+1}{3}\implies \stackrel{improper}{\cfrac{4}{3}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{5}{8}\div 1\frac{1}{3}\implies \cfrac{5}{8}\div \cfrac{4}{3}\implies \cfrac{5}{8}\cdot \cfrac{3}{4}\implies \cfrac{15}{32}[/tex]

Step-by-step explanation:

First consider the given line 8x = 2y - 5. This can be rewritten as

2y = 8x + 5

y = (8x + 5)/2 = 4x + 2.5

The line is y = 4x + 2.5. This is of the form y = mx + b.

On comparing, we see that the slope of the given line is m = 4

Now, we require to find the slope of a line perpendicular to this. Let its slope be m'.

We know that for two lines to be perpendicular, the product of the slopes = -1

So, m * m' = -1

4 * m' = -1

m' = -1/4

Answer: Slope of the perpendicular line is -1/4.

Step-by-step explanation:

82÷100= 0.82

0.82×164= 134.48 /134

Answer:

inding the Greatest Common Factor (GCF): To find the GCF of two expressions:

Factor each coefficient into primes. Write all variables with exponents in expanded form.

List all factors—matching common factors in a column. ...

Bring down the common factors that all expressions share.

Multiply the factors

Answer:

Step-by-step explanation:

2x = 2*x

4y = 2 *2 *y

6z = 2* 3 * z

GCF = 2

GCF is the greatest common factor. Here 2 is common factor

2x - 4y + 6z = 2*(x - 2y + 3z)

Answer:

Focus is (0,64)

Step-by-step explanation:

Directrix is y = -64

Axis of Symmetry is x = 0

Answer:

y = -2/7(x) + 3

Step-by-step explanation:

The equation of a line can be stated as y = mx + b, where m is the slope and b is the y-intercept (the value of the function when x = 0).  To find the equation of the line, we'll start by finding the slope.

The slope can be expressed as [tex]\frac{y_{2} - y_{1}}{x_{2}-{x_1}}[/tex]

Substituting in our coordinates, we have:

[tex]m = \frac{1 - 5}{7 - (-7)} = \frac{-4}{7 + 7} = \frac{-4}{14}[/tex], which can be simplified to [tex]-\frac{2}{7}[/tex]

Plugging that back into our equation, we have y = [tex]-\frac{2}{7}[/tex]x + b

Now, to find b, we substitute in one of our sets of coordinates.  Let's use (-7, 5)

x = -7 and y = 5, which gives us:

[tex]5 = -\frac{2}{7} (-7) + b[/tex]

[tex]-\frac{2}{7} (-7) = 2[/tex], which gives us:

5 = 2 + b

Subtracting 2 from both sides, we get:

3 = b

Plugging that back into our equation, we have [tex]y = -\frac{2}{7} x + 3[/tex]

Answer:

21 pounds

tell me if it helped :P

Step-by-step explanation:

Answer:

The answer is A

Step-by-step explanation:

When you reflect something over the x-axis all of the negative y coordinates become positive.

Step-by-step explanation:

e^4x = 5

x = 1/4 × ln(5)

x ÷ 1/4 = ln(5)

x.4 = ln(5)

4x = ln(5)

ln(5) = 4x

Option → D

Answer:

yes

Step-by-step explanation:

you can use the vertical line test

9514 1404 393

Answer:

  d(32) = 28.80

  the price Marcus pays on an item with an original price of 32

Step-by-step explanation:

  d(32) = 32 -0.1(32)

  d(32) = 28.8

The problem statement tells you d(32) is the price Marcus pays when the original price is 32.

Answer:

0

Step-by-step explanation:

Because it is a line parallel to x-axis.

Answer:

Graph A

Step-by-step explanation:

Graph the parabola using the direction, vertex, focus, and axis of symmetry.

Direction: Opens Up

Vertex:  ( 0 ,  2 )

Focus:  

( 0 ,  41 /20 )

Axis of Symmetry:  x = 0

Directrix:  y = 39 /20

x                   y

− 2                22

− 1                 7

0                  2

1                    7

2                  22

Answer:

8/10

Step-by-step explanation:

Answer:

the answer will be choice D

as we can see that the triangle have angles measuring 30 60 and 90 so we will use the 30-60-90 rule

you can look up the explantion about the rule online

Step-by-step explanation:

this is a kind of trick question, actually.

with whatever we draw, we produce X values as power of 3.

to be precise, we can have only

3⁰ = 1

3¹ = 3

3² = 9

3⁴ = 81

due to the possible combinations of drawn numbers (e.g. 3 cannot be created by a multiplication of 0s, 1s and 2s).

so, mostly, these results cannot be exact factors of 1024.

1024 cannot be divided by 3, nor by 9 nor by 81.

but 1024 is a multiple of 1 (as is every number).

so, we are looking at the probability to get 0 as multiplication result of the numbers on the 2 drawn balls.

the only possibilities are

1 and 0

2 and 0

out of in total 6 (2×3) different outcomes

1 and 0

1 and 1

1 and 2

2 and 0

2 and 1

2 and 2

the probability of this "0" event is again

number of desired outcomes / number of possible outcomes = 2/6 = 1/3

Answer:

all of the expressions are equivalent

Step-by-step explanation:

if you simplify each expression, you get the same answer for all of them

Answer:

3

Step-by-step explanation:

The slope is 3. It's the correct answer.

Answer:

9 players rushed 20 or more yards this season

Step-by-step explanation:

Answer:

1.4.5

2.idont know the answer for that question because I am grade 6