Final answer:
The formula for the nth derivative of the function f(x) = xe^7x involves the product of powers of x, powers of 7, and the exponential function e^7x as determined by applying Leibniz's rule iteratively.
Step-by-step explanation:
The student has asked to find a formula for the nth derivative of the function f(x) = xe7x, for any positive integer n. The general method to find the nth derivative of a product of functions involves using Leibniz's rule, which is a generalization of the product rule for derivatives.
To find the nth derivative, we apply Leibniz's rule iteratively, understanding that e7x is an exponential function, and its derivatives will be of the form 7ke7x, where k is the number of derivatives taken. The application of this rule combined with the derivative of the polynomial part x will provide the general form of the nth derivative.
For the nth derivative, we can express it as a sum involving the terms x and e7x, with n defined coefficients and powers of 7 corresponding to the derivative level. We will end up with terms that are products of powers of x, powers of 7, and the exponential function e7x.