Understanding the Limit Definition of a Derivative: Calculating Instantaneous Rate of Change in Calculus With Examples

Limit Definition of Derivative

The limit definition of a derivative is a way to calculate the rate of change or the instantaneous rate of change at a specific point on a graph

The limit definition of a derivative is a way to calculate the rate of change or the instantaneous rate of change at a specific point on a graph. It provides the foundation for understanding and calculating derivatives in calculus.

Let’s say we have a function f(x), and we want to find the derivative of this function at a certain point x=a. The limit definition of the derivative is described by the following equation:

f'(a) = lim (h→0) [f(a+h) – f(a)] / h

In this equation, h represents a small change in the x-value from the point of interest (x=a) and approaches zero. By taking the limit of this equation as h approaches zero, we can obtain the derivative of the function at the point x=a.

To understand this concept more clearly, let’s work through an example:

Example:
Consider the function f(x) = x^2. We want to find the derivative of this function at the point x=3, using the limit definition.

First, let’s substitute the values into the limit definition equation:

f'(3) = lim (h→0) [f(3+h) – f(3)] / h

Next, we can evaluate this expression:

f'(3) = lim (h→0) [(3+h)^2 – 3^2] / h
= lim (h→0) [(9 + 6h + h^2) – 9] / h
= lim (h→0) (6h + h^2) / h
= lim (h→0) (h(6 + h)) / h

Now, we can cancel out the h’s:

f'(3) = lim (h→0) (6 + h)

Finally, we substitute h=0 into the expression:

f'(3) = 6

So, the derivative of the function f(x) = x^2, at the point x=3, is 6.

This is the basic idea of the limit definition of derivative. By taking the limit as h approaches zero, we can calculate the slope of the curve at a particular point and determine the rate of change of the function at that point.

More Answers: