## Instantaneous Rate of Change at x=a

### The instantaneous rate of change at x=a represents the rate at which a function is changing at a specific point, represented by the value a

The instantaneous rate of change at x=a represents the rate at which a function is changing at a specific point, represented by the value a. It is also known as the derivative of the function evaluated at x=a.

To calculate the instantaneous rate of change at x=a, we can use the concept of the derivative. The derivative of a function f(x) with respect to x can be represented as f'(x) or dy/dx. So, the instantaneous rate of change at x=a can be calculated by finding the derivative of the function and evaluating it at x=a.

Here is the step-by-step process to find the instantaneous rate of change at x=a:

1. Identify the function: Start by identifying the function for which you want to find the instantaneous rate of change.

2. Determine the derivative: Use the rules of differentiation to find the derivative of the function. The derivative indicates how the function is changing at different points.

3. Evaluate the derivative at x=a: Once you have the derivative function, substitute x=a into it. This will give you the instantaneous rate of change at x=a.

For example, let’s say we have the function f(x) = 2x^2 – 5x + 3 and we want to find the instantaneous rate of change at x=2.

1. The function is f(x) = 2x^2 – 5x + 3.

2. Calculate the derivative:

f'(x) = 4x – 5

3. Substitute x=2 into f'(x):

f'(2) = 4(2) – 5 = 3

Therefore, the instantaneous rate of change at x=2 for the function f(x) = 2x^2 – 5x + 3 is 3. This means that at x=2, the function is changing at a rate of 3 units per unit increase in x.

It’s important to note that the instantaneous rate of change provides information about how the function is changing at a specific point, and it can be used for a variety of applications in calculus and real-world problem-solving.

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