Let \( F(x)=f(f(x)) \) and \( G(x)=(F(x))^{2} \) You also know that \( f(3)=8, f(8)=3, f^{\prime}(8)=11, f^{\prime}(3)=10 \) Find \( F^{\prime}(3)=\square \) and \( G^{\prime}(3)= \)

To find \( F'(3) \), we start with the function \( F(x) = f(f(x)) \). We'll use the chain rule to differentiate: \[ F'(x) = f'(f(x)) \cdot f'(x) \] Now, substituting \( x = 3 \): \[ F'(3) = f'(f(3)) \cdot f'(3) \] From the information given, we have \( f(3) = 8 \), so: \[ F'(3) = f'(8) \cdot f'(3) \] We know \( f'(8) = 11 \) and \( f'(3) = 10 \). Plugging in these values: \[ F'(3) = 11 \cdot 10 = 110 \] Next, we need to find \( G'(3) \). Given \( G(x) = (F(x))^2 \), we can apply the chain rule again: \[ G'(x) = 2F(x) \cdot F'(x) \] Now substituting \( x = 3 \): \[ G'(3) = 2F(3) \cdot F'(3) \] We already found \( F(3) = f(f(3)) = f(8) = 3 \), hence: \[ G'(3) = 2 \cdot 3 \cdot F'(3) = 2 \cdot 3 \cdot 110 = 660 \] Thus, we have: \[ F'(3) = 110 \quad \text{and} \quad G'(3) = 660 \]