Use the Chain Rule to differentiate the following function. \[ f(x)=\sqrt{x^{2}+\sqrt{2-5 x^{3}}} \] \[ f^{\prime}(x)= \]

Step 1 ：Given the function \(f(x)=\sqrt{x^{2}+\sqrt{2-5 x^{3}}}\)

Step 2 ：We need to find its derivative using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 3 ：In this case, the outer function is the square root function and the inner function is \(x^{2}+\sqrt{2-5 x^{3}}\).

Step 4 ：First, we find the derivative of the inner function, which itself is a composite function. The derivative of \(x^{2}+\sqrt{2-5 x^{3}}\) is \(-15x^{2}/(2\sqrt{2 - 5x^{3}}) + 2x\).

Step 5 ：Next, we find the derivative of the outer function, which is the square root function. The derivative of the square root of a function is \(1/(2\sqrt{function})\).

Step 6 ：We then multiply this by the derivative of the inner function to get the derivative of the composite function. The derivative of the function \(f(x)=\sqrt{x^{2}+\sqrt{2-5 x^{3}}}\) is \((-15x^{2}/(4\sqrt{2 - 5x^{3}}) + x)/\sqrt{x^{2}+\sqrt{2-5x^{3}}}\).

Step 7 ：\(\boxed{f^\prime(x)=\frac{-15x^{2}}{4\sqrt{2-5x^{3}}} + \frac{x}{\sqrt{x^{2}+\sqrt{2-5x^{3}}}}}\)