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 y shift with respect to x 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 ?

Answer:

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?

Answer:

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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