$$f'(x)=0$$

In mathematics, a derivative measures how a function's output changes in response to small changes in its input. It represents the rate of change or slope of the function at a specific point. Denoted as \(f'(x)\) or \(\frac{dy}{dx}\), it provides crucial information about the function's instantaneous rate of change, enabling us to analyze functions, find tangent lines, and solve optimization problems. Essentially, derivatives help us understand how a function behaves locally, making them a fundamental concept in calculus.