Final answer:
The nth derivative of the function f(x) = x * e^{7x} can be found using Leibniz's rule and is given by f^{(n)}(x) = x * 7^n * e^{7x} + n * 7^{n-1} * e^{7x}.
Step-by-step explanation:
To find a formula for the nth derivative of the function f(x) = x * e^{7x}, we can use Leibniz's rule for the product of two functions. Given the formulas for the derivatives of e^{7x} and x, which are 7^n * e^{7x} and 1 (or 0 for all derivatives higher than the first), respectively, we can sum over all the ways to choose i derivatives from x and n-i derivatives from e^{7x}. For the nth derivative of the product, we thus have:
f^{(n)}(x) = \sum_{i=0}^{n} {n \choose i} \frac{d^i}{dx^i}(x) \frac{d^{n-i}}{dx^{n-i}}(e^{7x})
More explicitly, this becomes:
f^{(n)}(x) = x \cdot 7^n \cdot e^{7x} + n \cdot 7^{n-1} \cdot e^{7x} since the higher-order derivatives of x vanish after the first differentiation (x becoming 1 and then 0 for all subsequent derivatives).