Understanding the Limit Definition of a Derivative: Calculating Instantaneous Rate of Change with the Limit Definition

Limit Definition of a Derivative at a Point

The limit definition of a derivative at a point is a fundamental concept in calculus that helps us understand the concept of instantaneous rate of change of a function

The limit definition of a derivative at a point is a fundamental concept in calculus that helps us understand the concept of instantaneous rate of change of a function.

Let’s say we have a function f(x) and we want to find the derivative of this function at a specific point x=a. The derivative at x=a is denoted as f'(a) or dy/dx evaluated at x=a.

The limit definition of a derivative at a point is given by the formula:

f'(a) = lim(h->0) (f(a+h) – f(a))/h

where f(a+h) represents the value of the function at the point a+h, f(a) represents the value of the function at the point a, and h represents a small change in x (approaching 0).

To find the derivative at x=a, we take the limit of the difference quotient as h approaches 0. This gives us the slope of the tangent line to the curve of the function at the point x=a.

Let’s understand this with an example:
Consider the function f(x) = x^2. We want to find the derivative at x=2 using the limit definition.

Step 1: Substitute the values into the formula:
f'(2) = lim(h->0) (f(2+h) – f(2))/h

Step 2: Calculate the values of f(2+h) and f(2):
f(2+h) = (2+h)^2 = 4 + 4h + h^2
f(2) = 2^2 = 4

Step 3: Substitute the values into the limit definition:
f'(2) = lim(h->0) [(4 + 4h + h^2) – 4]/h

Step 4: Simplify the expression:
f'(2) = lim(h->0) (4h + h^2)/h
f'(2) = lim(h->0) (h(4 + h))/h
f'(2) = lim(h->0) (4 + h)

Step 5: Evaluate the limit:
f'(2) = 4

Therefore, the derivative of f(x) = x^2 at x=2 is f'(2) = 4.

By using the limit definition of a derivative, we can determine the rate at which a function is changing at a specific point. This concept is crucial for understanding calculus and its applications in various fields such as physics, engineering, and economics.

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