Below is a table containing information about differentiable functions f and g:

x: -2 -1 0 1 2

f(x): 4 3 5 2 1

f'(x):1 4 2 5 3

g(x): 5 2 1 3 4

g'(x):3 5 4 1 2

F(x) + g(x), G(x)=f(x) * g(x), H(x) = g(x)/f(x)
Find F'(-1)
Find G'(0)
Find H'(-2)

G = fg

G' = f'g + fg', so
G'(0) = (2)(1)+(5)(4) = 22

similarly for the others, using the appropriate rules.

To find F'(-1), we need to calculate the derivative of the sum of F(x) and g(x) with respect to x, and evaluate it at x = -1.

Given that:
F(x) = f(x) + g(x)

We can find F'(-1) by using the following steps:

Step 1: Calculate f'(-1) and g'(-1) using the respective derivative functions:
f'(-1) = 4
g'(-1) = 5

Step 2: Add f'(-1) and g'(-1):
F'(-1) = f'(-1) + g'(-1)
F'(-1) = 4 + 5
F'(-1) = 9

Therefore, F'(-1) = 9.

To find G'(0), we need to calculate the derivative of the product of f(x) and g(x) with respect to x, and evaluate it at x = 0.

Given that:
G(x) = f(x) * g(x)

We can find G'(0) by using the following steps:

Step 1: Calculate f'(0) and g'(0) using the respective derivative functions:
f'(0) = 2
g'(0) = 4

Step 2: Multiply f(0) and g'(0):
G'(0) = f(0) * g'(0)
G'(0) = 5 * 4
G'(0) = 20

Therefore, G'(0) = 20.

To find H'(-2), we need to calculate the derivative of the quotient of g(x) and f(x) with respect to x, and evaluate it at x = -2.

Given that:
H(x) = g(x) / f(x)

We can find H'(-2) by using the following steps:

Step 1: Calculate f(-2) and g'(-2) using the respective derivative functions:
f(-2) = 3
g'(-2) = 1

Step 2: Multiply f(-2) and g'(-2):
H'(-2) = g'(-2) * f(-2)
H'(-2) = 1 * 3
H'(-2) = 3

Therefore, H'(-2) = 3.

To find F'(-1), we need to find the derivative of F(x) = f(x) + g(x) and then evaluate it at x = -1.

The derivative of the sum of two functions is the sum of their derivatives. Therefore, F'(x) = f'(x) + g'(x).

We are given the values of f'(x) and g'(x) in the table:

f'(x): 1 4 2 5 3
g'(x): 3 5 4 1 2

To find F'(-1), we need to find the sum of the corresponding derivatives at x = -1:

F'(-1) = f'(-1) + g'(-1)

To find f'(-1) and g'(-1), we can look at the given table:

x: -2 -1 0 1 2
f'(x): 1 4 2 5 3
g'(x): 3 5 4 1 2

From the table, we can see that f'(-1) = 4 and g'(-1) = 5.

Therefore, F'(-1) = f'(-1) + g'(-1) = 4 + 5 = 9.

To find G'(0), we need to find the derivative of G(x) = f(x) * g(x) and then evaluate it at x = 0.

The derivative of the product of two functions is given by the product rule: (fg)' = f'g + fg'.

We are given the values of f'(x) and g'(x) in the table:

f'(x): 1 4 2 5 3
g'(x): 3 5 4 1 2

To find G'(0), we need to find the product of f'(0) and g(0) and then evaluate it at x = 0:

G'(0) = f'(0)g(0) + f(0)g'(0)

From the table, we can see that f(0) = 5 and g(0) = 1.

G'(0) = f'(0)g(0) + f(0)g'(0) = 2 * 1 + 5 * 4 = 2 + 20 = 22.

To find H'(-2), we need to find the derivative of H(x) = g(x) / f(x) and then evaluate it at x = -2.

The derivative of the quotient of two functions is given by the quotient rule: (f/g)' = (f'g - fg') / g^2.

We are given the values of f'(x), f(x), g'(x), and g(x) in the table:

f'(x): 1 4 2 5 3
f(x): 4 3 5 2 1
g'(x): 3 5 4 1 2
g(x): 5 2 1 3 4

To find H'(-2), we need to substitute the given values into the quotient rule:

H'(-2) = (g'(-2)f(-2) - g(-2)f'(-2)) / (f(-2))^2

From the table, we can see that f(-2) = 5, f'(-2) = 2, g(-2) = 5, and g'(-2) = 3.

H'(-2) = (g'(-2)f(-2) - g(-2)f'(-2)) / (f(-2))^2
= (3*5 - 5*2) / 5^2
= (15 - 10) / 25
= 5 / 25
= 1/5

Therefore, H'(-2) = 1/5.