Final answer:
The alternative definition of a derivative, known as the geometric definition, focuses on the slope of the tangent line to the graph of a function at a specific point. This is different from the traditional definition, which involves taking the limit of a difference quotient. An example illustrating the geometric definition is the function f(x) = x^2, where the derivative at x = 2 is 4.
Step-by-step explanation:
The alternative definition of a derivative is often referred to as the geometric definition. In this definition, the derivative of a function is the slope of the tangent line to the graph of the function at a particular point. This definition focuses on the idea of the rate of change of the function at a specific point, rather than the limit concept used in the traditional definition.
For example, let's consider the function f(x) = x^2. Using the geometric definition of the derivative, we can find the derivative of f(x) by drawing a tangent line to the graph of f(x) at a given point, say x = 2. The slope of this tangent line represents the derivative of f(x) at x = 2, which is equal to 4. This means that the rate of change of f(x) at x = 2 is 4.
In contrast, the traditional definition of the derivative involves taking the limit of a difference quotient as the change in x approaches 0. It focuses on the idea of instantaneous rate of change and provides a general formula for finding the derivative of a function, rather than computing it at a specific point. The traditional definition states that the derivative of a function f(x) is given by the limit as h approaches 0 of [f(x + h) - f(x)]/h.
For the same function f(x) = x^2, using the traditional definition of the derivative, we can find that f'(x) = 2x. This equation represents the derivative of f(x) at any point x, not just at a specific point like in the geometric definition.