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If the log base is "e" ....it's more simple......but still a little messy

5th derivative of   f(x)   =  sin x * ln (x)

 It will be the same answer as before except  that we will not have any logs in the denominators :

d^5/dx^5(sin(x) log(x)) = (24 sin(x))/x^5 - (30 cos(x))/x^4 - (20 sin(x))/x^3 + (10 cos(x))/x^2 + (5 sin(x))/x + log(x) cos(x)

See Wolfram's computation, here :

https://www.wolframalpha.com/input/?i=fifth+derivative+%5B+sin+x+*+ln+(x)+%5D

Find f(11) for f(x) = [x^2-9] / [27-x^3]     factoring first may reduce the work a little

[ (x + 3) (x - 3) ]  /  [  (3 - x ) ( 9 + 3x  + x^2 ) ]  =      factor a negative 1 out of the second factor in the numerator

- [ ( x + 3) (3 - x) ]  / [  (3 - x ) ( 9 + 3x  + x^2 ) ]    =  (3 -x)  "cancels"

- ( x + 3)   / ( x^2 + 3x + 9)

So f(11)   =

- (11 + 3) / ( 11^2 + 33 + 9 )  =

- (14) / (121 + 33 + 9 )  =

- (14) / ( 163)

Nicely done, Max !  Thanx !

-25 + (-9)   =

- 34        we are just adding negatives....

One year  =   60 seconds * 60 minutes * 24 hours  * 365 days   =  31,536,000 seconds

So....

1,000,000,000 sec  / 31,536,000 sec   ≈  31.7  years

Upshot : you will be lucky to live 3 billion seconds........and nobody has lived 4 billion seconds  [ yet ]

So I expected.

e^i pi

I do not see a question here ://

or if it is log base e

\(y=e^xsinx\\~\\ y'=e^xsinx+e^xcosx\\~\\ y''=e^xsinx+e^xcosx+e^xcosx-e^xsinx\\ y''=2e^xcosx\\~\\ y'''=2e^xcosx-2e^xsinx\\ y'''=2e^x(cosx-sinx)\\ y''''=2e^x(cosx-sinx)+2e^x(-sinx-cosx)\\ y''''=2e^x(cosx-sinx-sinx-cosx)\\ y''''=-4e^xsinx\\ y'''''=-4y'=-4e^x(sinx+cosx)\\ \)

Thanks Chris,

It was a slightly unusual question, I liked it too :)

Very nice, Melody...I like this one...!!!

\(\sin^2 x - \sin x = 2\\ \sin^2 x - \sin x - 2 = 0 \\\text{Let a = sin x}\\ a^2 - a - 2 = 0\\ (a-2)(a+1) = 0\\ (\sin x - 2)(\sin x + 1) = 0\\ \sin x = 2\text{(rejected) or }\sin x = -1\\ \sin x = -1\\ x = \dfrac{3\pi}{2} \text{rad}\)

\(\cos^2 x + \cos x = \sin^2x\\ \cos^2x + \cos x = 1 - \cos^2x\\ 2\cos^2x + \cos x - 1 = 0\\ \text{Let a = cos x this time.}\\ 2a^2 + a - 1 = 0\\ (2a - 1)(a + 1) = 0\\ \cos x = \dfrac{1}{2} \text{ or }\cos x = -1\\ x = \dfrac{\pi}{3}\text{ rad} \text{ or }x = \pi \text{ rad}\)

Rauhan:

Your answer for 1 is correct, it is water.

I am not too sure for Question 2...... But my answer is different with yours.

Not sure how to prove it, but just by looking at it 'a' is 3/2 pi

Hi guest :)

John has a empty box. He puts some red counters and some blue counters. The ratio of the number of red counters to blue counters is 1:4. Linda takes at random 2 counters. The probability that she took 2 red counters is 6/155. How many red counters did John put in the box?

Let there be x red counters, then there will be 4x blue counters which equals 5x counters altogether in the box.

so the probability that the first counter will be red is     \(\frac{x}{5x}=\frac{1}{5}\)

Now there are x-1 red counters left out of   5x-1 counters in total so 

the probablility that the second counter is red is    \(\frac{x-1}{5x-1}\)

\(P( both\; red) = \frac{1}{5}\times \frac{(x-1)}{(5x-1)}=\frac{6}{155}\\~\\ \begin{align} \frac{(x-1)}{5(5x-1)}&=\frac{6}{155}\\ (x-1)*155&=6*5(5x-1)\\ 155x-155&=150x-30\\ 5x&=125\\ x&=25\\ \end{align}\)

So there was 25 red marbles in the box :)

77,500,000,000 in scientific notaion?

=7.75 x 10^10

∆ = b² - 4 a c -b± √∆ x1,2 = ———— 2a

 y=15-3x

 0=15-3x   |+3x

3x=15       | :3

  x=5

2- - (-2) =2+2=4

asinus without login

20 power 0.78

\(20^{ 0.78}=10.34673668\)     ( Directly on computer )

\(=e^{ln\ 20\times 0.78 }=e^{2.9957\times 0.78}\)

\(=e^{2.33667117337}\)

\(=10.3467366799\)

  !

7867

And just so u know I am not a nerd.

1) water

2) anger

1 mile = 1760 yds....so 0.5 mile  =  880 yds

So....we can use a proportion to find out how long - in x minutes - it will take you to walk 880 yds

20 / 1408  =   x / 880      cross-multiply

880 * 20  = 1408 * x

17600  = 1408 * x       divide both sides by 1408

12.5 min  =  x = time to walk  0.5 miles

Since you want to get there 10 min early,  you need to be there at 12:01

And 12.5 min before this =  11:48: 30 AM  is the time you should leave your house

x=5

Let N  be the number of rocks in the bag....and we have that

N + 100,000,000   - 36  = 10,000N  - 5N   simplify

N + 100,000,000 - 36  =  9995N

N + 99,999,964  =  9995N

99,999,964  = 9994N     divide both sides by 9994

10,006 = N   = the number of small rocks in Melvin's bag