Answer
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Work Step by Step
The derivative plays a fundamental role in defining slopes, tangents, and rates of change in calculus. Specifically:
1. Slopes: The derivative of a function at a given point represents the slope of the tangent line to the graph of the function at that point. By calculating the derivative at various points, you can determine how the slope of the function changes as you move along its graph.
2. Tangents: The derivative also helps define the equation of the tangent line to a curve at a specific point. The slope of this tangent line is given by the derivative at that point. This is crucial for understanding the behavior of functions at specific points and for approximating the behavior of the function near those points.
3. Rates of Change: In real-world applications, derivatives are used to represent rates of change. For instance, if you have a function representing the position of an object over time, its derivative would represent the object's velocity at any given time. Similarly, the derivative of a function representing the amount of a substance in a chemical reaction with respect to time would represent the rate of change of that substance.
In summary, the derivative provides a powerful tool for analyzing the behavior of functions, determining slopes, finding tangent lines, and understanding rates of change in various contexts.
