Understanding the Limit Definition of Derivative | An Essential Concept in Calculus

Limit Definition of Derivative

The limit definition of a derivative is a mathematical expression that defines the derivative of a function at a specific point

The limit definition of a derivative is a mathematical expression that defines the derivative of a function at a specific point. It is expressed using the limit operator.

Let’s consider a function f(x) and a specific value a. The derivative of f(x) at a is denoted as f'(a) or dy/dx|a, and can be defined using the limit as follows:

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

In this expression, h represents a very small change in the x-value around a. By taking the limit of this expression as h approaches 0, we determine the instantaneous rate of change, or the slope, of the function at the point (a, f(a)). This slope represents the derivative of the function at that specific point.

To further understand this definition, let’s consider an example. Suppose we have the function f(x) = x^2 and we want to find the derivative f'(2) at the point x = 2. We can apply the limit definition:

f'(2) = lim (h->0) [(2+h)^2 – 2^2] / h
= lim (h->0) [(4 + 4h + h^2) – 4] / h
= lim (h->0) (4h + h^2) / h
= lim (h->0) (4 + h)
= 4

Hence, the derivative of f(x) = x^2 at x = 2 is equal to 4.

The limit definition of derivative is fundamental in calculus as it allows us to find the instantaneous rate of change of a function at any given point. It serves as the basis for many derivative rules and formulas used in calculus.

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