Find the derivative of the function. \[ f(x)=4 e^{x}-x^{5 e} \] \( f^{1}(x)=4 e^{x}-5 x^{4} e \)

Absolutely, let’s spice things up with some fascinating context! Did you know that the concept of derivatives was independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century? Newton used it in his work on motion and change while Leibniz formulated it in a more systematic way which eventually led to the notation we use today, like \( f'(x) \) for derivatives. Their collaboration through competition helped lay the groundwork for modern calculus, making it a thrilling period of mathematical discovery! Speaking of applications, derivatives aren't just academic exercises; they have real-world implications! Engineers use derivatives to analyze the changing speeds and accelerations of objects, economists apply them to find maximum profit or minimum cost, and physicists employ them to describe the motion of particles. Essentially, derivatives help us understand how things change, making them essential tools in science, business, and beyond!