The **relative rate of change (RROC) **is the ratio of a function’s derivative to itself.

The RROC of *y* = *f*(*t*) at *t *= *a *is defined as [1]:

Relative rates of change are often expressed as the percentage change of y per unit change in x; for example, if the amount in an investment increases in value from $1000 to $1400 dollars over ten years, then the amount increases at an average rate of 4 percent per year during that ten-year interval.

## Relative Rate of Change: Examples

**Example question #1**: What is the relative rate of change of f(x) = x^{3} at x = 4?

**Solution**: The derivative of f(x) = x^{3} is 3x^{2} (using the power rule), so:

RROC = f′(4) / f(4) = 3 * 4^{2} / 4^{3} = 48 / 64 = 3/4

**Example question #2:** Corn production in the U.S., C = f(t), in millions of tones, is a function of *t* years since the beginning of 2010. If f(8) = 253 and f′(8) = 17, what is the RROC of C at t = 8? Interpret it in terms of corn production.

Solution: RROC = f′(8) / f(8) = 17/253 ≈ 0.0672 = 6.72%

In 2018, corn production in the U.S. grew by 6.72% per year.

**Example question #2:** Annual world production of photovoltaic (PV) cells, S, in megawatts, is approximated by S = 277e^{0.368t}, where *t* is years since 2000. Estimate the RROC of PV cell production in 2010 using Δ*t* = 1.

**Solution**:

We’re told that S = 277e^{0.368t} is an *estimate*.

We’re given Δ*t* = 1, we have to format the RROC equation to include that fact.

Step 1: Find the derivative at t = 10 (i.e. 2010):

f′(10) = f(11) – f(10) / 11 – 10 = 277e^{0.368(11)} – 277e^{0.368(10)} / 1

= 15867.33 – 10982.05

= 4885.28.

Step 2: Find RROC. We already know f(10) from Step 1, so:

RROC = f′(10) / f(10)

= 4885.28 / 10982.05

= .44484 or 44.484%.

## References

[1] Math 124. 2: Rate of Change: The derivative. Retrieved July 3, 2021 from: https://users.math.msu.edu/users/liuqinbo/chapter2.pdf

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