1. The problem asks: What does it mean to find the derivative at an arbitrary value? 2. The derivative of a function $f(x)$ at a point $x=a$ is defined as the limit of the average rate of change of the function as the interval approaches zero: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ 3. This formula means we are calculating the slope of the tangent line to the curve $y=f(x)$ at the point $x=a$. 4. In simpler terms, finding the derivative at $x=a$ tells us how fast the function is changing exactly at that point. 5. It is like zooming in infinitely close to the curve at $x=a$ and measuring the steepness of the curve there. 6. This instantaneous rate of change is different from average rate of change over an interval because it considers an infinitely small interval. 7. So, when we find the derivative at an arbitrary value, we are finding the exact slope of the function at that point, which tells us the function's behavior locally.