Find and simplify the derivative of the following functions.
f(x) = √(e^2x + 8x²e^x +16x⁴) (Hint: Factor the function under the square root first.)

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = \sqrt{e^{2x} + 8x^2e^x + 16x^4} is a composition of functions, where the outer function is the square root and the inner function is a polynomial expression. To simplify the differentiation process, factor the expression under the square root.
2. Step 2: Notice that the expression under the square root, e^{2x} + 8x^2e^x + 16x^4, can be factored. Look for a common factor or a pattern that resembles a perfect square trinomial. In this case, it can be factored as (e^x + 4x^2)^2.
3. Step 3: Substitute the factored form back into the original function: f(x) = \sqrt{(e^x + 4x^2)^2}. The square root and the square cancel each other out, simplifying the function to f(x) = e^x + 4x^2.
4. Step 4: Differentiate the simplified function f(x) = e^x + 4x^2. Use the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives. Differentiate each term separately: the derivative of e^x is e^x, and the derivative of 4x^2 is 8x.
5. Step 5: Combine the derivatives of each term to find the derivative of the entire function: f'(x) = e^x + 8x. This is the simplified derivative of the original function.