Question

Let F(x)= f(f(x)) and G(x)=(F(x))^{2} and suppose that f(3)=12, f(12)=2, f'(12)=5, f'(3)=10Find F'(3) and G'(3).F'(3)= ?

Answer 1

To find F'(3), differentiate F(x) = f(f(x)) using the chain rule. F'(3) = f'(f(3)) * f'(3). To find G'(3), differentiate G(x) = (F(x))^2 using the chain rule. G'(3) = 2 * F(3) * F'(3).

Step-by-step explanation:

To find F'(3), we need to differentiate the function F(x) = f(f(x)).

We know that f(3) = 12 and f'(3) = 10. So, to find F'(3), we can apply the chain rule:

1. Differentiate the outer function: F'(x) = f'(f(x))
2. Multiply by the derivative of the inner function: f'(x)

Substituting in x = 3, we have F'(3) = f'(f(3)) * f'(3) = f'(12) * f'(3). From the given information, f'(12) = 5.

To find G'(3), we need to differentiate the function G(x) = (F(x))^2. Using the chain rule, we find G'(x) = 2 * F(x) * F'(x). Substituting in x = 3, we can calculate G'(3) = 2 * F(3) * F'(3). From the given information, F(3) = f(f(3)) = f(12) = 2.