When we measure a rate of change at a specific instant in time, then it is called an instantaneous rate of change. On the other hand, the average rate of change will tell about the average rate at which some term was changing over some period of time. While we are on the way to the grocery store, then the speed will be constantly changing. Sometimes we may be moving faster than 20 km per hour and sometimes slower. At each instant of time, the instantaneous rate of change will correspond to the speed at that exact moment. In this article, we will discuss the instantaneous rate of change formula with examples. Let us begin learning!

Source: socratic.org

**Instantaneous Rate of Change Formula**

**What is the Instantaneous Rate of Change?**

The rate of change at one known instant or point of time is the Instantaneous rate of change. It is equivalent to the value of the derivative at that specific point of time. Therefore, we can say that, in a function, the slope m of the tangent will give the instantaneous rate of change at a specific

**The formula for Instantaneous Rate of Change:**

The average rate of change of variable y with respect to the variable x is the difference quotient. Now if we look at the difference quotient and let us suppose \( \Delta x tending to zero.\)

This will give the instantaneous rate of change. It must be noted that the time interval gets lesser and lesser.

Therefore Instantaneous Rate of Change Formula provided with limit exists in,

**f'(x)=\( \lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}\)**

**i.e. f'(a)=\( \lim_{h \rightarrow 0}\frac{f(a + h)-f(a)}{h} \)**

When y = f(x), with regards to x, when x = a.

For example, we can compute the Instantaneous Speed Formula as below:

Speed is the rate of change of position of some object with respect to time. The speed of an object may change as it moves. Thus the instantaneous speed is the speed of an object at a certain instant of time. If we represent the position as a function of time, then the speed will depend on the change in the position as time changes.

Hence, the instantaneous speed can be found as this change in time becomes small. To calculate the instantaneous speed we need to find the limit of the position function as the change in time approaches zero.

**Solved Examples**

Q.1: Compute the Instantaneous rate of change of the some function f(x) given as,

f(x)= 3x² + 12 at x = 5 ?

Solution:

The given function is,

f(x) = 3x² + 12

Finding its derivative with respect to x,

f'(x) = 3(2x) + 0

f’(x) = 6x

Therefore, the instantaneous rate of change at point x = 5 will be,

f'(5) = 6 × 5

f’(5) = 30

Thus solution is 30.

Q.2: Compute the Instantaneous rate of change of the function given as:

f(x) = 5x³ – 4x² + 2x + 1 at x = 0 ?

Answer:

Given function is:

f(x) = 5x³ – 4x² + 2x + 1

After differentiation,

f'(x) = 5(3x²) – 4(2x) + 2 + 0

f’(x) = 15x² – 8x + 2

Therefore, the instantaneous rate of change at x = 0 will be

**f'(0) = 2**

**Thus solution is 2.**