Calculating the Instantaneous Rate of Change: A Step-by-Step Guide with Example

Instantaneous Rate of Change at x=a

The instantaneous rate of change of a function at a particular point, denoted as x = a, represents the rate at which the function is changing at that specific value of x

The instantaneous rate of change of a function at a particular point, denoted as x = a, represents the rate at which the function is changing at that specific value of x.

To determine the instantaneous rate of change at x = a, you need to calculate the derivative of the function and evaluate it at x = a.

Step 1: Find the derivative of the function f(x) using the appropriate differentiation rules. If you are given the function explicitly, differentiate it using power rule, chain rule, product rule, or quotient rule, depending on the form of the function. If you are given the function implicitly, you can differentiate both sides of the equation and solve for dy/dx.

Step 2: Simplify the derivative obtained in Step 1 if necessary.

Step 3: Evaluate the derivative at x = a by substituting a into the derived function. This will give you the instantaneous rate of change at that specific point.

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

Step 1: Differentiate the function.

f'(x) = 2x – 3

Step 2: The derivative of the function is already simplified, so no additional simplification is needed.

Step 3: Evaluate the derivative at x = 2.

f'(2) = 2(2) – 3
= 4 – 3
= 1

Therefore, the instantaneous rate of change of the function f(x) = x^2 – 3x + 2 at x = 2 is 1.

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