Question

Let F(x)=f(f(x)) and G(x)=(F(x))²and suppose that f(9)=10,f(10)=2,f ′ (10)=12,f ′ (9)=5 Find F ′(9) and G ′ (9)

Answer 1

The derivative of F at 9, F'(9), is 60. The derivative of G at 9, G'(9), is 240. This is found using the chain rule and plugging in the given functions and derivatives.

Step-by-step explanation:

To find F'(9), we first need to use the chain rule. The chain rule states that if you have a composed function like F(x) = f(f(x)), then when you differentiate it, you do so by taking the derivative of the outside function evaluated at the inside function times the derivative of the inside function. So, we have:

F'(x) = f'(f(x)) · f'(x).

Plugging in the given values:

F'(9) = f'(f(9)) · f'(9) = f'(10) · f'(9) = 12 · 5 = 60.

For G'(9), where G(x) = (F(x))^2, we also use the chain rule. We differentiate the outer function, which is the square function, and multiply it by the derivative of the inside function, F(x):

G'(x) = 2 · F(x) · F'(x).

Given that F(9) = f(f(9)) = f(10) = 2, we substitute to get:

G'(9) = 2 · F(9) · F'(9) = 2 · 2 · 60 = 240.