The Limit Definition of the Derivative: Understanding Instantaneous Rate of Change in Calculus

Limit Definition of Derivative

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

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

Let’s consider a function f(x) and a specific point x=a. The derivative of f(x) at x=a, denoted as f'(a) or dy/dx|a, represents the instantaneous rate of change of f(x) at x=a. It gives us the slope of the tangent line to the graph of f(x) at that point.

The limit definition of the derivative is expressed as:

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

In this expression, h represents a small change in the x-values around the point x=a. By taking the limit as h approaches 0, we can determine the exact rate of change at that point.

To compute the derivative using the limit definition, follow these steps:

Step 1: Set up the expression.
Begin by writing down the limit definition of the derivative:

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

Step 2: Simplify the expression.
Expand the numerator and simplify the expression as much as possible using algebraic techniques. Combine like terms and factor out common terms if necessary.

Step 3: Take the limit.
Substitute h = 0 into the expression and compute the limit to evaluate the derivative.

Here’s an example to illustrate the process:

Consider the function f(x) = 2x^2 + 3x – 1. We want to find the instantaneous rate of change at x=1, i.e., f'(1).

Step 1: Set up the expression.
f'(1) = lim(h->0) [f(1 + h) – f(1)] / h

Step 2: Simplify the expression.
Let’s substitute the values into the expression:
f'(1) = lim(h->0) [2(1+h)^2 + 3(1+h) – 1 – (2(1)^2 + 3(1) – 1)] / h

Expanding and simplifying:
f'(1) = lim(h->0) [2(1+2h+h^2) + 3 + 3h – 1 – 2 – 3] / h
= lim(h->0) [2h^2 + 4h + 2] / h

Step 3: Take the limit.
Let’s simplify further and substitute h = 0 into the expression:
f'(1) = lim(h->0) 2h + 4 + 2/h

As h approaches 0, the last term 2/h approaches infinity. Therefore, the derivative is undefined at x = 1.

By following the steps of the limit definition of the derivative, we can find the instantaneous rate of change at any given point on a function. This concept is essential in further studying calculus and understanding the behavior of functions.

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