Fill in the blank.The of a function f at x represents the slope of the graph of f at the point (x, f(x)).

The derivative of a function $f$ at $x$ represents the slope of the graph of $f$ at the point $(x, f(x))$.

The derivative is a fundamental concept in calculus and is denoted by $f'(x)$ or $\frac{df}{dx}$. It provides information about how the function $f$ changes as the input $x$ varies. Specifically, the derivative measures the instantaneous rate of change of the function at a specific point, indicating how the function is increasing or decreasing at that point.

For example, if the derivative $f'(x)$ is positive, the graph of the function is increasing at $x$, whereas if $f'(x)$ is negative, the function is decreasing. If $f'(x) = 0$, it may indicate a local maximum, minimum, or a point of inflection. Understanding derivatives is crucial for analyzing functions' behavior, optimizing processes, and solving real-world problems in various fields such as physics, economics, and engineering.