( (a) Let $f(x)=\sqrt{9+8 x^{3}}$. Find $f^{\prime}(x)$. \[ f^{\prime}(x)= \] (b) Let $f(x)=e^{\sqrt{9+8 x^{3}}}$. Find $f^{\prime}(x)$. \[ f^{\prime}(x)= \]

Step 1 ：Identify the outer function as \(\sqrt{u}\) and its derivative as \(\frac{1}{2\sqrt{u}}\)

Step 2 ：Identify the inner function as \(9+8x^3\) and its derivative as \(24x^2\)

Step 3 ：Apply the chain rule to find the derivative of \(f(x)=\sqrt{9+8 x^{3}}\), which is \(f^{\prime}(x) = \frac{1}{2\sqrt{9+8x^3}} \cdot 24x^2\)

Step 4 ：Simplify the derivative to get \(f^{\prime}(x) = \frac{12x^2}{\sqrt{9+8x^3}}\)

Step 5 ：\(\boxed{f^{\prime}(x) = \frac{12x^2}{\sqrt{9+8x^3}}}\) is the derivative of \(f(x)=\sqrt{9+8 x^{3}}\)

Step 6 ：Identify the outer function as \(e^u\) and its derivative as \(e^u\)

Step 7 ：Identify the inner function as \(\sqrt{9+8x^3}\) and its derivative as \(\frac{12x^2}{\sqrt{9+8x^3}}\)

Step 8 ：Apply the chain rule to find the derivative of \(f(x)=e^{\sqrt{9+8 x^{3}}}\), which is \(f^{\prime}(x) = e^{\sqrt{9+8x^3}} \cdot \frac{12x^2}{\sqrt{9+8x^3}}\)

Step 9 ：Simplify the derivative to get \(f^{\prime}(x) = \frac{12x^2 e^{\sqrt{9+8x^3}}}{\sqrt{9+8x^3}}\)

Step 10 ：\(\boxed{f^{\prime}(x) = \frac{12x^2 e^{\sqrt{9+8x^3}}}{\sqrt{9+8x^3}}}\) is the derivative of \(f(x)=e^{\sqrt{9+8 x^{3}}}\)