Find 100th Derivative of Cubic Exponential Function at Zero

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

When tasked with finding the 100th derivative of a function like $f(x) = x^3 e^{2x}$ at a specific point, such as $x = 0$, it is essential to leverage the power of calculus and the rules governing derivatives. In this specific instance, the function is a product of a polynomial $x^3$ and an exponential function $e^{2x}$, each having unique derivative patterns. Knowing the derivative patterns of exponential functions and polynomials will be invaluable. The exponential function $e^{2x}$ has a derivative that involves extracting the constant growth factor of 2 with each differentiation, while the polynomial will decrease in order until it is reduced to zero.

A strategic approach to this problem is the application of Taylor series expansion, which provides a way to express complex functions as an infinite sum of their derivatives at a single point. The Taylor series expansion for the exponential function is particularly straightforward given its infinite differentiability. Moreover, because we're evaluating at $x = 0$, the task simplifies into finding how these derivatives interact at zero, focusing particularly on non-zero coefficients of the power series expansion. Recognizing the superposition of these series will simplify the computational process to yield the specified derivative at zero, which often involves identifying terms that vanish and those that contribute to the final result.

In summary, tackling a problem that involves such high-order derivatives requires both an understanding of calculus rules and computational strategies. Applying Taylor or Maclaurin series is a robust method when multiple derivatives are involved, especially around specific points like zero, as it transforms the infinite derivative sequence into a manageable algebraic problem.