Answer:
A Swedish court said Friday that rapper ASAP Rocky will be detained for another week as prosecutors decide whether to formally charge him following a fight last month in downtown Stockholm.
The rapper, born Rakim Mayers, has been in Swedish custody, along with two of his associates, since early July after a June 30 altercation that Rocky’s supporters say he tried to avoid. Under the court’s decision Friday, the men could be detained until at least July 25.
Concerns have mounted for the men amid reports that they are being held in “inhumane conditions,” including long daily periods of solitary confinement. A growing number of lawmakers and celebrities, including Kim Kardashian and Kanye West, has called for their release.
Answer:
See Below.
Step-by-step explanation:
We have the equation:
![\displaystyle y = \left(3e^{2x}-4x+1\right)^{{}^1\! / \! {}_2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20y%20%3D%20%5Cleft%283e%5E%7B2x%7D-4x%2B1%5Cright%29%5E%7B%7B%7D%5E1%5C%21%20%2F%20%5C%21%20%7B%7D_2%7D)
And we want to show that:
![\displaystyle y \frac{d^2y }{dx^2} + \left(\frac{dy}{dx}\right) ^2 = 6e^{2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%5Cfrac%7Bd%5E2y%20%7D%7Bdx%5E2%7D%20%2B%20%5Cleft%28%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright%29%20%5E2%20%3D%206e%5E%7B2x%7D)
Instead of differentiating directly, we can first square both sides:
![\displaystyle y^2 = 3e^{2x} -4x + 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5E2%20%3D%203e%5E%7B2x%7D%20-4x%20%2B%201)
We can find the first derivative through implicit differentiation:
![\displaystyle 2y \frac{dy}{dx} = 6e^{2x} -4](https://tex.z-dn.net/?f=%5Cdisplaystyle%202y%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%206e%5E%7B2x%7D%20-4)
Hence:
![\displaystyle \frac{dy}{dx} = \frac{3e^{2x} -2}{y}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B3e%5E%7B2x%7D%20-2%7D%7By%7D)
And we can find the second derivative by using the quotient rule:
![\displaystyle \begin{aligned}\frac{d^2y}{dx^2} & = \frac{(3e^{2x}-2)'(y)-(3e^{2x}-2)(y)'}{(y)^2}\\ \\ &= \frac{6ye^{2x}-\left(3e^{2x}-2\right)\left(\dfrac{dy}{dx}\right)}{y^2} \\ \\ &=\frac{6ye^{2x} -\left(3e^{2x} -2\right)\left(\dfrac{3e^{2x}-2}{y}\right)}{y^2}\\ \\ &=\frac{6y^2e^{2x}-\left(3e^{2x}-2\right)^2}{y^3}\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%26%20%3D%20%5Cfrac%7B%283e%5E%7B2x%7D-2%29%27%28y%29-%283e%5E%7B2x%7D-2%29%28y%29%27%7D%7B%28y%29%5E2%7D%5C%5C%20%5C%5C%20%26%3D%20%5Cfrac%7B6ye%5E%7B2x%7D-%5Cleft%283e%5E%7B2x%7D-2%5Cright%29%5Cleft%28%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%29%7D%7By%5E2%7D%20%5C%5C%20%5C%5C%20%26%3D%5Cfrac%7B6ye%5E%7B2x%7D%20-%5Cleft%283e%5E%7B2x%7D%20-2%5Cright%29%5Cleft%28%5Cdfrac%7B3e%5E%7B2x%7D-2%7D%7By%7D%5Cright%29%7D%7By%5E2%7D%5C%5C%20%5C%5C%20%26%3D%5Cfrac%7B6y%5E2e%5E%7B2x%7D-%5Cleft%283e%5E%7B2x%7D-2%5Cright%29%5E2%7D%7By%5E3%7D%5Cend%7Baligned%7D)
Substitute:
![\displaystyle y\left(\frac{6y^2e^{2x}-\left(3e^{2x}-2\right)^2}{y^3}\right) + \left(\frac{3e^{2x}-2}{y}\right)^2 =6e^{2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5Cleft%28%5Cfrac%7B6y%5E2e%5E%7B2x%7D-%5Cleft%283e%5E%7B2x%7D-2%5Cright%29%5E2%7D%7By%5E3%7D%5Cright%29%20%2B%20%5Cleft%28%5Cfrac%7B3e%5E%7B2x%7D-2%7D%7By%7D%5Cright%29%5E2%20%3D6e%5E%7B2x%7D)
Simplify:
![\displaystyle \frac{6y^2e^{2x}- \left(3e^{2x} -2\right)^2}{y^2} + \frac{\left(3e^{2x}-2\right)^2}{y^2}= 6e^{2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B6y%5E2e%5E%7B2x%7D-%20%5Cleft%283e%5E%7B2x%7D%20-2%5Cright%29%5E2%7D%7By%5E2%7D%20%2B%20%5Cfrac%7B%5Cleft%283e%5E%7B2x%7D-2%5Cright%29%5E2%7D%7By%5E2%7D%3D%206e%5E%7B2x%7D)
Combine fractions:
![\displaystyle \frac{\left(6y^2e^{2x}-\left(3e^{2x} - 2\right)^2\right) +\left(\left(3e^{2x}-2\right)^2\right)}{y^2} = 6e^{2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cleft%286y%5E2e%5E%7B2x%7D-%5Cleft%283e%5E%7B2x%7D%20-%202%5Cright%29%5E2%5Cright%29%20%2B%5Cleft%28%5Cleft%283e%5E%7B2x%7D-2%5Cright%29%5E2%5Cright%29%7D%7By%5E2%7D%20%3D%206e%5E%7B2x%7D)
Simplify:
![\displaystyle \frac{6y^2e^{2x}}{y^2} = 6e^{2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B6y%5E2e%5E%7B2x%7D%7D%7By%5E2%7D%20%3D%206e%5E%7B2x%7D)
Simplify:
![6e^{2x} \stackrel{\checkmark}{=} 6e^{2x}](https://tex.z-dn.net/?f=6e%5E%7B2x%7D%20%5Cstackrel%7B%5Ccheckmark%7D%7B%3D%7D%206e%5E%7B2x%7D)
Q.E.D.
Answer:
<
Step-by-step explanation:
![\dfrac{4}{9}=0.\overline{4}](https://tex.z-dn.net/?f=%5Cdfrac%7B4%7D%7B9%7D%3D0.%5Coverline%7B4%7D%3C0.45)
The tenths-place digits of the two numbers are the same, 4. So comparison proceeds to the hundredths place. The digit in the hundredths place of 0.4444... is 4, which is less than the digit in the hundredths place of 0.45, which is 5. Hence 4/9 < 0.45.
Answer:
k would be 55
Step-by-step explanation:
35+90=125 and assuming that you're looking for a supplementary angle, the total must equal 180 so subtract 180-125 and you get k=55
That would be c 4x^3-x^2+3x+13