Find \( \begin{array}{l}f^{\prime}(x) \\ f(x)=5 x e^{x} \\ f^{\prime}(x)=5 e^{x}+5 x e^{x}\end{array} \)

Let’s kick it off with some historical context! The product rule of differentiation, which is used here, was developed in the 17th century by mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz. It's fascinating how their work laid the groundwork for calculus, allowing us to differentiate functions efficiently and understand change! Now, let's dive into the real-world applications! Differentiating functions like \( f(x) = 5xe^x \) is crucial in fields such as physics, engineering, and economics. For instance, this function could represent a growth model where the population increases exponentially with respect to some resource. Calculating \( f'(x) \) can help determine growth rates, optimize processes, or analyze trends over time—it’s all about giving you that edge in understanding dynamic systems!