Differentiate the following function. \[ f(x)=x^{6} e^{4 x} \] \[ f^{\prime}(x)= \]

Step 1 ：Given the function \(f(x)=x^{6} e^{4 x}\)

Step 2 ：We need to differentiate this function.

Step 3 ：To do this, we use the product rule and the chain rule.

Step 4 ：The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 5 ：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 6 ：In this case, we can consider \(x^{6}\) as the first function and \(e^{4 x}\) as the second function.

Step 7 ：The derivative of \(x^{6}\) is \(6x^{5}\) and the derivative of \(e^{4 x}\) is \(4e^{4 x}\) (using the chain rule).

Step 8 ：So, the derivative of the function \(f(x)=x^{6} e^{4 x}\) is \(f^{\prime}(x)=6x^{5}e^{4 x} + 4x^{6}e^{4 x}\).

Step 9 ：Final Answer: The derivative of the function \(f(x)=x^{6} e^{4 x}\) is \(\boxed{4x^{6}e^{4 x} + 6x^{5}e^{4 x}}\)