The prime factorization is 3*3*3*2 or 3^3 * 2
Differentiate both sides of
![6x f(x)^4 + 9x^3 f(x) = 78](https://tex.z-dn.net/?f=6x%20f%28x%29%5E4%20%2B%209x%5E3%20f%28x%29%20%3D%2078)
with respect to x :
![\left(6x f(x)^4 + 9x^3 f(x)\right)' = (78)' \\\\ 6\left(xf(x)^4\right)' + 9\left(x^3f(x)\right)' = 0](https://tex.z-dn.net/?f=%5Cleft%286x%20f%28x%29%5E4%20%2B%209x%5E3%20f%28x%29%5Cright%29%27%20%3D%20%2878%29%27%20%5C%5C%5C%5C%206%5Cleft%28xf%28x%29%5E4%5Cright%29%27%20%2B%209%5Cleft%28x%5E3f%28x%29%5Cright%29%27%20%3D%200)
Use the product and chain rules for the left side:
![\left(xf(x)^4\right)' = (x)'f(x)^4 + x\left(f(x)^4\right)' = f(x)^4 + 4xf(x)^3f'(x)](https://tex.z-dn.net/?f=%5Cleft%28xf%28x%29%5E4%5Cright%29%27%20%3D%20%28x%29%27f%28x%29%5E4%20%2B%20x%5Cleft%28f%28x%29%5E4%5Cright%29%27%20%3D%20f%28x%29%5E4%20%2B%204xf%28x%29%5E3f%27%28x%29)
![\left(x^3f(x)\right)' = \left(x^3\right)'f(x) + x^3f'(x) = 3x^2f(x)+x^3f'(x)](https://tex.z-dn.net/?f=%5Cleft%28x%5E3f%28x%29%5Cright%29%27%20%3D%20%5Cleft%28x%5E3%5Cright%29%27f%28x%29%20%2B%20x%5E3f%27%28x%29%20%3D%203x%5E2f%28x%29%2Bx%5E3f%27%28x%29)
Solve for f '(x) :
![6\left(f(x)^4+4xf(x)^3f'(x)\right) + 9\left(3x^2f(x)+x^3f'(x)\right) = 0 \\\\ 6f(x)^4+27x^2f(x) + \left(24xf(x)^3+9x^3\right)f'(x) = 0 \\\\ f'(x) = -\dfrac{6f(x)^4+27x^2f(x)}{24xf(x)^3+9x^3}](https://tex.z-dn.net/?f=6%5Cleft%28f%28x%29%5E4%2B4xf%28x%29%5E3f%27%28x%29%5Cright%29%20%2B%209%5Cleft%283x%5E2f%28x%29%2Bx%5E3f%27%28x%29%5Cright%29%20%3D%200%20%5C%5C%5C%5C%206f%28x%29%5E4%2B27x%5E2f%28x%29%20%2B%20%5Cleft%2824xf%28x%29%5E3%2B9x%5E3%5Cright%29f%27%28x%29%20%3D%200%20%5C%5C%5C%5C%20f%27%28x%29%20%3D%20-%5Cdfrac%7B6f%28x%29%5E4%2B27x%5E2f%28x%29%7D%7B24xf%28x%29%5E3%2B9x%5E3%7D)
Then
![f'(1) = \boxed{\dfrac{14}{61}}](https://tex.z-dn.net/?f=f%27%281%29%20%3D%20%5Cboxed%7B%5Cdfrac%7B14%7D%7B61%7D%7D)
It would take 7 hours to get to New York City
A table would look something like this:
0 1 2 3 4 5 6 7
0 50 100 150 200 250 300 350
Hope you can read it and this helps
Answer:
D. 6/15
Step-by-step explanation:
You divide 6 (the number of times landed on a specific place) by 15 (the number of times spun). When you solve for 6/15, the answer is .4. In the form as a percentage, it would be 40%.
Answer:
The answer is D. y-10=2(x-1)
Step-by-step explanation:
![y-y_{1}=m(x-x_{1})](https://tex.z-dn.net/?f=y-y_%7B1%7D%3Dm%28x-x_%7B1%7D%29)
First, find the slope from the two points:
m=(10-2)/(1+3)=2
let ![y_{1}=10 and x_{1}=1](https://tex.z-dn.net/?f=y_%7B1%7D%3D10%20and%20x_%7B1%7D%3D1)
thus, y-10=2(x-1)