Final answer:
The nth derivative of the function f(x) = 4e^(2x) is found by applying the chain rule and constant multiple rule repeatedly, resulting in the formula f^(n)(x) = 4 × (2^n)e^(2x), where n is the order of the derivative.
Step-by-step explanation:
To find a formula for the nth derivative of the function f(x) = 4e^(2x), we can use the fact that the derivative of e^(ax), where a is a constant, is ae^(ax). Since we're dealing with the derivative of an exponential function multiplied by a constant, we can apply the constant multiple rule and the chain rule of differentiation successively.
Let's denote the nth derivative of f(x) as f(n)(x). The first derivative of f(x) would be f'(x) = 4 × 2e^(2x), since the derivative of e^(2x) is 2e^(2x). Similarly, the nth derivative will involve taking the derivative n times, each time multiplying by the constant 2 which comes from the exponent.
Therefore, the nth derivative of f(x) is given by f(n)(x) = 4 × (2^n)e^(2x), where n is the number of times the derivative has been taken.