## definition of derivative f'(x)

### The derivative of a function, denoted as f'(x), is a concept in calculus that measures the rate at which a function is changing at a specific point

The derivative of a function, denoted as f'(x), is a concept in calculus that measures the rate at which a function is changing at a specific point. It represents the slope of the tangent line to the graph of the function at that specific point.

Mathematically, the derivative of a function f(x) is defined as the limit of the difference quotient as the change in x approaches zero:

f'(x) = lim(h -> 0) [(f(x + h) – f(x))/h]

This means that to find the derivative of a function, we take the difference in y-values of the function for a small change in x, and divide it by that small change in x. Then, we take the limit of this expression as the change in x approaches zero.

The derivative captures the instantaneous rate of change of the function at a particular point. It tells us how fast the function is increasing or decreasing at that point.

Geometrically, the derivative represents the slope of the tangent line to the graph of the function. If the derivative is positive, the function is increasing at that point, and if it is negative, the function is decreasing. A derivative of zero indicates a horizontal tangent line.

The derivative has various applications in mathematics and real-life scenarios. It helps us analyze the behavior of functions, find critical points, determine maximum and minimum values, and solve optimization problems. Additionally, it plays a crucial role in calculus, as it is used to find integrals and solve differential equations.

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