# Let us find the instantaneous rate of change for different functions

**Instantaneous rate of change calculus **gives the change in rate at a given instant.** **Before proceeding further, let us **define rate of change in math. **It is given by the change of a quantity between two instances.

The **instantaneous rate of change of a function **gives the derivative of the same function at particular point according to the **instantaneous rate of change definition**. The **formula for instantaneous rate of change** will help to calculate the same for any given function.

Consider a function f(x) for which we will be **finding instantaneous rate of change **over the interval (a,b). We have Δf = f(b) – f(a) and Δx = b – a. Hence the **average rate of change over an interval **(a,b) will be while the **instantaneous rate of change **will be This **instantaneous rate of change formula **is similar to that of differentiation. Thus we have seen the **calculus instantaneous rate of change **for the function. So its time to move on with related examples.

## Evaluation of instantaneous rate of change examples

The **instantaneous rate of change examples **given below will help to have a profound knowledge of the topic. So it is advised to study them seriously.

(1) **Find the instantaneous rate of change **for y = x at x = 4:

Here y = x, Δy = f(x) – f(4) = x – 4 and Δx = x – 4. So Δy/Δx = 1 => the limit value is also 1. Thus the rate of change for above equation is 1 irrespective of the points.

(2) **Find instantaneous rate of change **for y = x^{2} – x + 1 at x = 3:

We have Δy = f(x) – f(3) = (x^{2} – x + 1) – (3^{2} – 3 + 1) = x^{2} – x + 1 – 7 = x^{2} – x – 6. Factorizing the above equation, Δy = (x + 2)(x – 3). Now Δx = (x – 3) since we need to find it for x = 3. So the ratios will be Δy/Δx = (x + 2)(x – 3) / (x – 3) = (x + 2). Therefore R.O.C_{inst} = [Δy / Δx]_{x->3} = [x + 2]_{x->3} = 3 + 2 = 5.

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