Let \( F(x)=f\left(x^{4}\right) \) and \( G(x)=(f(x))^{4} \) You also know that \( a^{3}=11, f(a)=3, f^{\prime}(a)=4, f^{\prime}\left(a^{4}\right)=13 \) Then \( F^{\prime}(a)=\square \) and \( G^{\prime}(a)= \)

To find \( F^{\prime}(a) \) and \( G^{\prime}(a) \), we need to use the chain rule and the power rule for differentiation. Starting with \( F(x) = f(x^4) \): Using the chain rule, we have: \[ F^{\prime}(x) = f^{\prime}(x^4) \cdot (x^4)^{\prime} = f^{\prime}(x^4) \cdot 4x^3 \] Now, substituting \( x = a \) gives: \[ F^{\prime}(a) = f^{\prime}(a^4) \cdot 4a^3 \] We know: - \( a^3 = 11 \) so \( 4a^3 = 4 \times 11 = 44 \) - \( f^{\prime}(a^4) = 13 \) Thus, \[ F^{\prime}(a) = 13 \cdot 44 = 572 \] Now, for \( G(x) = (f(x))^4 \): Using the chain rule: \[ G^{\prime}(x) = 4(f(x))^3 \cdot f^{\prime}(x) \] Substituting \( x = a \): \[ G^{\prime}(a) = 4(f(a))^3 \cdot f^{\prime}(a) \] We know: - \( f(a) = 3 \) so \( (f(a))^3 = 3^3 = 27 \) - \( f^{\prime}(a) = 4 \) Thus, \[ G^{\prime}(a) = 4 \cdot 27 \cdot 4 = 432 \] Final answers: \[ F^{\prime}(a) = 572 \] \[ G^{\prime}(a) = 432 \]