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3 years ago

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Out of a group of 3 people, 2 study French and 1 studies Spanish. Two people are drawn randomly from the group without replaceme

nt. Display all possible outcomes as an organized list.

1 answer:

zimovet [89]3 years ago

7 0

Answer:

- 1 French, 1 Spanish

- 2 French

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Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

mote1985 [20]

Answer:

$\frac{d}{dx}\left(\ln \left(\frac{x}{x^2+1}\right)\right)=\left(\ln{\left(\frac{x}{x^{2} + 1} \right)}\right)^{\prime }=\frac{-x^2+1}{x\left(x^2+1\right)}$

Step-by-step explanation:

To find the derivative of the function $y(x)=\ln \left(\frac{x}{x^2+1}\right)$ you must:

Step 1. Rewrite the logarithm:

$\left(\ln{\left(\frac{x}{x^{2} + 1} \right)}\right)^{\prime }=\left(\ln{\left(x \right)} - \ln{\left(x^{2} + 1 \right)}\right)^{\prime }$

Step 2. The derivative of a sum is the sum of derivatives:

$\left(\ln{\left(x \right)} - \ln{\left(x^{2} + 1 \right)}\right)^{\prime }}={\left(\left(\ln{\left(x \right)}\right)^{\prime } - \left(\ln{\left(x^{2} + 1 \right)}\right)^{\prime }\right)$

Step 3. The derivative of natural logarithm is $\left(\ln{\left(x \right)}\right)^{\prime }=\frac{1}{x}$

${\left(\ln{\left(x \right)}\right)^{\prime }} - \left(\ln{\left(x^{2} + 1 \right)}\right)^{\prime }={\frac{1}{x}} - \left(\ln{\left(x^{2} + 1 \right)}\right)^{\prime }$

Step 4. The function $\ln{\left(x^{2} + 1 \right)}$ is the composition $f\left(g\left(x\right)\right)$ of two functions $f\left(u\right)=\ln{\left(u \right)}$ and $u=g\left(x\right)=x^{2} + 1$

Step 5. Apply the chain rule $\left(f\left(g\left(x\right)\right)\right)^{\prime }=\frac{d}{du}\left(f\left(u\right)\right) \cdot \left(g\left(x\right)\right)^{\prime }$

$-{\left(\ln{\left(x^{2} + 1 \right)}\right)^{\prime }} + \frac{1}{x}=- {\frac{d}{du}\left(\ln{\left(u \right)}\right) \frac{d}{dx}\left(x^{2} + 1\right)} + \frac{1}{x}\\\\- {\frac{d}{du}\left(\ln{\left(u \right)}\right)} \frac{d}{dx}\left(x^{2} + 1\right) + \frac{1}{x}=- {\frac{1}{u}} \frac{d}{dx}\left(x^{2} + 1\right) + \frac{1}{x}$

Return to the old variable:

$- \frac{1}{{u}} \frac{d}{dx}\left(x^{2} + 1\right) + \frac{1}{x}=- \frac{\frac{d}{dx}\left(x^{2} + 1\right)}{{\left(x^{2} + 1\right)}} + \frac{1}{x}$

The derivative of a sum is the sum of derivatives:

$- \frac{{\frac{d}{dx}\left(x^{2} + 1\right)}}{x^{2} + 1} + \frac{1}{x}=- \frac{{\left(\frac{d}{dx}\left(1\right) + \frac{d}{dx}\left(x^{2}\right)\right)}}{x^{2} + 1} + \frac{1}{x}=\frac{1}{x^{3} + x} \left(x^{2} - x \left(\frac{d}{dx}\left(1\right) + \frac{d}{dx}\left(x^{2}\right)\right) + 1\right)$

Step 6. Apply the power rule $\frac{d}{dx}\left(x^{n}\right)=n\cdot x^{-1+n}$

$\frac{1}{x^{3} + x} \left(x^{2} - x \left({\frac{d}{dx}\left(x^{2}\right)} + \frac{d}{dx}\left(1\right)\right) + 1\right)=\\\\\frac{1}{x^{3} + x} \left(x^{2} - x \left({\left(2 x^{-1 + 2}\right)} + \frac{d}{dx}\left(1\right)\right) + 1\right)=\\\\\frac{1}{x^{3} + x} \left(- x^{2} - x \frac{d}{dx}\left(1\right) + 1\right)\\$

$\frac{1}{x^{3} + x} \left(- x^{2} - x {\frac{d}{dx}\left(1\right)} + 1\right)=\\\\\frac{1}{x^{3} + x} \left(- x^{2} - x {\left(0\right)} + 1\right)=\\\\\frac{1 - x^{2}}{x \left(x^{2} + 1\right)}$

Thus, $\frac{d}{dx}\left(\ln \left(\frac{x}{x^2+1}\right)\right)=\left(\ln{\left(\frac{x}{x^{2} + 1} \right)}\right)^{\prime }=\frac{-x^2+1}{x\left(x^2+1\right)}$

3 0

3 years ago

Tan x + tan 2x = 0

choli [55]

stated method in question is wrong!

<u>Step-by-step explanation:</u>

We need to find the general solution of x ! for the equation $tan x + tan 2x = 0$

⇒ $tan x + tan 2x = 0$

⇒ $(tan x + tan 2x)-tan2x = 0-tan2x$

⇒ $tan x = -tan2x$

Now , we know that $tan(\pi -2x) = -tan2x$

⇒ $tan x = -tan2x$

⇒ $tan x = tan(\pi -2x)$

Now , we know that $TanA = TanB \\A=B$

⇒ $tan x = tan(\pi -2x)$

⇒ $x = \pi -2x$

⇒ $3x = \pi$

⇒ $x = \frac{\pi}{3}$ ......(1)

General solution of Tan x is : $x = n\pi + \alpha$ , \

Here , from equation (1) we get $\alpha =\frac{\pi }{3}$ . Hence general solution is :

⇒ $x = n\pi +\frac{\pi }{3}$ .Therefore , stated method in question is wrong!

4 0

3 years ago

Victor is ordering pizzas for a party. He would like to have 1/4 of a pizza for each guest. He can only order whole pizzas, not

sertanlavr [38]

7 pizzas, because you would divide 27 by 4 and get 6.75 and then round it up to 7.

Hope this helped :)

5 0

3 years ago

Read 2 more answers

Need the answer now pls

Schach [20]

Answer:

35x-35y

Step by step explanation: I

0.5(20x-50y+36)-0.25(100x+40y-12)

10x-25y+18+25x-10y+3 -------- after we distribute 0.5&-0.25

10x+25x-25y-10y+18+3. ---------we collect the like terms to add them or substract

35x-35y+21 -----------------------------simplified form

6 0

3 years ago

Read 2 more answers

Can someone give me the answers please?

lutik1710 [3]

Answer:

6. slope= -4/5 Y-intercept is 3

7.

a.y=-2x

b.y=1x+1

c.5/2x-1

d.y=4x+3

e.y=-3/2x-5

Step-by-step explanation:

3 0

3 years ago