## Formal definition of derivative

### The formal definition of derivative is the fundamental concept in calculus that describes the rate at which a function changes

The formal definition of derivative is the fundamental concept in calculus that describes the rate at which a function changes. It provides a precise way of measuring how the value of a function is changing as the input (independent variable) changes.

Let’s consider a function f(x) that is defined on an interval containing a point a. The derivative of f(x) at a, denoted as f'(a) (read as “f prime of a”), is defined as follows:

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

Here, “lim(h→0)” represents the limit as h approaches zero. The expression [f(a + h) – f(a)] / h represents the average rate of change of f(x) over the interval (a, a+h).

Taking the limit as h approaches zero provides the instantaneous rate of change of f(x) at the point a. In other words, it gives us the slope of the tangent line to the graph of f(x) at the point (a, f(a)). This slope represents how quickly f(x) is changing at that particular point.

In summary, the formal definition of the derivative quantifies the rate of change of a function at a specific point. It allows us to analyze the behavior of functions, find slopes of curves, and solve various applications involving instantaneous rates of change.

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