Differentiate the function. \[ G(x)=\sqrt[3]{x^{4}+6 x} \]

Step 1 ：Given the function \(G(x)=\sqrt[3]{x^{4}+6 x}\)

Step 2 ：We need to find its derivative.

Step 3 ：To do this, we can use 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 4 ：In this case, the outer function is the cube root function and the inner function is \(x^{4}+6x\).

Step 5 ：Differentiating the outer function gives us \(\frac{1}{3}(x^{4}+6x)^{-\frac{2}{3}}\) and differentiating the inner function gives us \(4x^{3}+6\).

Step 6 ：Multiplying these two results together gives us the derivative of the function.

Step 7 ：So, the derivative of the function \(G(x)=\sqrt[3]{x^{4}+6 x}\) is \(\boxed{\frac{4x^{3}+2}{3\left(x^{4}+6x\right)^{\frac{2}{3}}}}\)