 # Instantaneous Rate of Change Formula

## Instantaneous Rate of Change

The instantaneous rate of change is the change in the rate at a particular instant, and it is same as the change in the derivative value at a specific point. For a graph, the instantaneous rate of change at a specific point is the same as the tangent line slope. That is, it is a curve slope.
Another way to better grasp this definition is with the differential quotient and limits. The average rate of shift with respect to is the quotient of difference.

The Formula of Instantaneous Rate of Change represented with limit exists in,

$$\begin{array}{l}f'(a) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{x\to 0}\frac{[f(a + h) – f(a)]}{h}\end{array}$$

With respect to x, when x=a and y = f(x)

### Solved Example

Problem 1: Compute the Instantaneous rate of change of the function f(x) = 3x2 + 12 at x = 4 ?

Known Function,

y = f(x) = 3x2 + 12

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

f'(x) =6x

Thus, the instantaneous rate of change at x = 4

f'(4) = 6(4)

f'(4) = 24

Problem  2: Compute the Instantaneous rate of change of the function f(x) = 5x3 – 4x2 + 2x + 1 at x = 2?

Known Function,

y = f(x) = 5x3 – 4x2 + 2x + 1

f'(x) = 5(3x2) – 4(2x) + 2 + 0

f'(x) = 15x2 – 8x + 2

Thus, the instantaneous rate of change at x = 2

f'(2) = 15(2)2 – 8(2) + 2 = 60 – 16 + 2 = 46

f'(2) = 46

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