Answer:
f'(x) = -1/(1 - Cos(x))
Step-by-step explanation:
The quotient rule for derivation is:
For f(x) = h(x)/k(x)
![f'(x) = \frac{h'(x)*k(x) - k'(x)*h(x)}{k^2(x)}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7Bh%27%28x%29%2Ak%28x%29%20-%20k%27%28x%29%2Ah%28x%29%7D%7Bk%5E2%28x%29%7D)
In this case, the function is:
f(x) = Sin(x)/(1 + Cos(x))
Then we have:
h(x) = Sin(x)
h'(x) = Cos(x)
And for the denominator:
k(x) = 1 - Cos(x)
k'(x) = -( -Sin(x)) = Sin(x)
Replacing these in the rule, we get:
![f'(x) = \frac{Cos(x)*(1 - Cos(x)) - Sin(x)*Sin(x)}{(1 - Cos(x))^2}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7BCos%28x%29%2A%281%20-%20Cos%28x%29%29%20-%20Sin%28x%29%2ASin%28x%29%7D%7B%281%20-%20Cos%28x%29%29%5E2%7D)
Now we can simplify that:
![f'(x) = \frac{Cos(x)*(1 - Cos(x)) - Sin(x)*Sin(x)}{(1 - Cos(x))^2} = \frac{Cos(x) - Cos^2(x) - Sin^2(x)}{(1 - Cos(x))^2}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7BCos%28x%29%2A%281%20-%20Cos%28x%29%29%20-%20Sin%28x%29%2ASin%28x%29%7D%7B%281%20-%20Cos%28x%29%29%5E2%7D%20%3D%20%5Cfrac%7BCos%28x%29%20-%20Cos%5E2%28x%29%20-%20Sin%5E2%28x%29%7D%7B%281%20-%20Cos%28x%29%29%5E2%7D)
And we know that:
cos^2(x) + sin^2(x) = 1
then:
![f'(x) = \frac{Cos(x)- 1}{(1 - Cos(x))^2} = - \frac{(1 - Cos(x))}{(1 - Cos(x))^2} = \frac{-1}{1 - Cos(x)}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7BCos%28x%29-%201%7D%7B%281%20-%20Cos%28x%29%29%5E2%7D%20%3D%20-%20%5Cfrac%7B%281%20-%20Cos%28x%29%29%7D%7B%281%20-%20Cos%28x%29%29%5E2%7D%20%3D%20%5Cfrac%7B-1%7D%7B1%20-%20Cos%28x%29%7D)
Answer:
n=11
Step-by-step explanation:
n+1=4(n-8)
11+1=4(11-8)
12=4(3)
12=12
Answer:
£38.48
Step-by-step explanation:
First, we have to find what 4% of 37 is, so our equation is:
37 x 0.04 = 1.48
Then we have to add that 4% to the original cost:
1.48 + 37 = 38.48
So the total price of the theatre ticket is £38.48
Answer: 81 units²
Step-by-step explanation: To solve this problem, remember that the formula for the area of a square is 4s.
Therefore, since the perimeter of the given square is 36, we have 36 = 4s and dividing both sides by 4, 9 = s.
Now, remember that the formula for the area of a square is s² and since the length of a side of the given square is 9, we have (9)² or 81.
So the area of a square that has a perimeter of 36 is 81 units².