Calculating the Derivative Using the Formal Definition: Understanding the Rate of Change and Tangent Line Slope

formal version of def. of derivative

The formal definition of the derivative of a function at a particular point can be expressed as follows:

Let f(x) be a function defined on an interval containing the point x = a

The formal definition of the derivative of a function at a particular point can be expressed as follows:

Let f(x) be a function defined on an interval containing the point x = a. The derivative of f(x) at x = a, denoted as f'(a), is defined by the limit:

f'(a) = lim (h→0) [f(a + h) – f(a)] / h

In this definition, h represents a small change in the x-value, known as the increment. The derivative f'(a) represents the instantaneous rate of change of the function at the point x = a, or the slope of the tangent line to the graph of the function at that point.

To compute the derivative using this formal definition, we evaluate the limit as h approaches zero. By finding this limit, we obtain the derivative of the function at that specific point. This definition allows us to determine the rate at which a function is changing at a particular point, providing valuable insights into its behavior and characteristics.

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