Understanding Instantaneous Rate of Change and its Relationship to Derivatives: Step-by-Step Explanation and Example

Given some function f the instantaneous rate of change at x=c if given by f'(. c)

To understand the concept of instantaneous rate of change and how it relates to the derivative of a function, let’s break it down step by step

To understand the concept of instantaneous rate of change and how it relates to the derivative of a function, let’s break it down step by step.

1. Instantaneous Rate of Change:
The instantaneous rate of change at a specific point on a graph represents how fast a function is changing at that particular point. It can be thought of as the slope of the tangent line to the graph of the function at that point. The instantaneous rate of change provides information about how the function’s output (y-values) is changing as its input (x-values) changes.

2. The Derivative of a Function:
The derivative of a function represents the rate at which the function is changing at any given instant. It measures the slope or rate of change of the function at each point on the graph. The derivative is denoted by f'(x) or dy/dx and it gives us a new function that describes the rate of change of the original function f(x).

3. Relating the Instantaneous Rate of Change to the Derivative:
When we talk about the instantaneous rate of change at a specific point x=c, we are essentially referring to the derivative of the function f(x) evaluated at that point. This can be expressed as f'(c). So, f'(c) gives us the instantaneous rate of change of the function f(x) at x=c.

To calculate f'(c), we need to find the derivative of the function f(x) using various differentiation rules such as power rule, product rule, chain rule, etc. Once we have the derivative function, we can plug in the value c to find the instantaneous rate of change at x=c.

Let’s consider an example to illustrate this:

Example:
Given the function f(x) = x^2 + 3x – 4, find the instantaneous rate of change at x=2.

Step 1: Calculate the derivative of f(x):
f'(x) = (d/dx)(x^2 + 3x – 4)

Using the power rule and the sum rule of derivatives, we get:
f'(x) = 2x + 3

Step 2: Evaluate f'(c) at x=2:
f'(2) = 2(2) + 3
f'(2) = 4 + 3
f'(2) = 7

Therefore, the instantaneous rate of change at x=2 for the function f(x) = x^2 + 3x – 4 is 7.

Remember, the derivative gives us the instantaneous rate of change of a function at any given point. So, f'(c) represents the instantaneous rate of change of the function f(x) at x=c.

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