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Differentiation is a fundamental concept in calculus, which is the mathematical study of change. In its most basic form, differentiation involves finding the rate at which a quantity is changing at a specific point. This is often referred to as the derivative of a function. The derivative of a function f(x) at a point x=a is denoted as f'(a) or dy/dx at x=a, and it represents the slope of the tangent line to the curve of the function at that point.

There are several methods to find the derivative of a function, depending on its form. For polynomial functions, the power rule is commonly used, which states that if f(x) = x^n, then f'(x) = nx^(n-1). For trigonometric functions, specific rules apply, such as the derivative of sine being cosine and vice versa.

Differentiation has numerous applications in various fields, including physics, engineering, economics, and biology. In physics, for example, differentiation is used to calculate velocity and acceleration from position and velocity functions, respectively. In economics, differentiation is employed to analyze the behavior of functions that model economic phenomena, such as cost, revenue, and profit functions.

The concept of differentiation can be extended to higher-order derivatives, which involve differentiating a function multiple times. The second derivative, for instance, represents the rate of change of the first derivative, and it is often used to analyze the concavity of a function. Higher-order derivatives can provide deeper insights into the behavior of functions and their applications in various fields.