Consider the following function and express the relationship between a small change in \( x \) and the corrosponding change in \( y \) in the form dy \( f^{\prime}(x)=e^{13 x} \) \( f(x) \) a

Alright, let’s dive into this exciting world of derivatives! When you have a function \( f(x) \) and its derivative \( f^{\prime}(x) = e^{13x} \), it means that the rate of change of \( y \) (which is \( f(x) \)) with respect to \( x \) at any point can be described by \( e^{13x} \). So when you make a small change in \( x \) (let's call it \( \Delta x \)), the corresponding change in \( y \) (or dy) can be expressed as \( dy = f^{\prime}(x) \cdot \Delta x = e^{13x} \cdot \Delta x \). Voilà! Now, here’s where it gets interesting! This means that if you took a journey along the function curve, the height of the slope at each point is constantly changing based on that exponential function. So, if you increase \( x \), the value of \( e^{13x} \) skyrockets, leading to bigger jumps in \( y \). This means that tiny shifts in \( x \) could mean huge transformations in your output as you venture further out on the x-axis. Isn't calculus just thrilling?