Differentiate the following function.
\[
f(x)=x^{6} e^{4 x}
\]
\[
f^{\prime}(x)=
\]
Final Answer: The derivative of the function \(f(x)=x^{6} e^{4 x}\) is \(\boxed{4x^{6}e^{4 x} + 6x^{5}e^{4 x}}\)
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 \(\boxed{4x^{6}e^{4 x} + 6x^{5}e^{4 x}}\)