100th Derivative of Polynomial Exponential Function at Zero

Find the 100th derivative of $f(x) = x^3 e^{2x}$ at $x = 0$.

This problem delves into the concept of finding higher-order derivatives, specifically the 100th derivative, of a product of functions, which in this case is a polynomial and an exponential function. It is an excellent demonstration of how series expansion, particularly the application of Taylor or Maclaurin series, can simplify the differentiation of functions that otherwise might seem quite complex to differentiate repeatedly.

The primary strategy to solve this problem involves recognizing that the function can be expressed as a product of a polynomial and an exponential. For such functions, repeated differentiation is efficiently managed by using the series expansion of the exponential function. The exponential function has a particularly convenient Maclaurin series expansion, which helps us derive results for repeated derivatives without directly applying derivative rules repeatedly for each order.

Besides, this problem is a practical application of the general formula for the coefficient of a power series term after successive differentiation, often involving the multinomial expansion of the product of series terms. Understanding this concept is crucial in subjects involving Taylor and Maclaurin series as it simplifies what could otherwise be a computationally intensive task. Furthermore, evaluating the derivative at a point $(x = 0)$, ties back to evaluating the Maclaurin series, where all terms other than the constant coefficient vanish, significantly simplifying the calculation.