Business Applications of Differentiation
2. Relative and Percentage Rates of Change
- Relative rate of change of a quantity $Q(x)$ with respect to $x$ is given by the ratio
$\left[\begin{array}{c}
\text { Relative rate } \
\text { of change }
\end{array}\right]=\frac{Q^{\prime}(x)}{Q(x)}=\frac{d Q / d x}{Q}$
- Percentage rate of change $Q(x)$ with respect to $x$ is
$\left[\begin{array}{c}
\text { Percentage relative rate } \
\text { of change }
\end{array}\right]=\frac{Q^{\prime}(x)}{Q(x)} \times 100
Answer (1)
Relative rate of change is defined as Q ( x ) Q ′ ( x ) .
Percentage rate of change is the relative rate of change multiplied by 100: Q ( x ) Q ′ ( x ) × 100 .
These rates indicate how a quantity changes relative to its current value.
The percentage rate provides a standardized measure for comparing growth or decline rates: Q ( x ) Q ′ ( x ) × 100 .
Explanation
Problem Analysis We are given the definitions of relative and percentage rates of change and are asked to restate them.
Relative Rate of Change The relative rate of change of a quantity Q ( x ) with respect to x is the ratio of the derivative of Q ( x ) to Q ( x ) itself. This is expressed as: Q ( x ) Q ′ ( x )
Percentage Rate of Change The percentage rate of change of a quantity Q ( x ) with respect to x is the relative rate of change multiplied by 100. This is expressed as: Q ( x ) Q ′ ( x ) × 100
Final Answer Therefore, the relative rate of change is Q ( x ) Q ′ ( x ) and the percentage rate of change is Q ( x ) Q ′ ( x ) × 100 .
Examples
In economics, if Q ( t ) represents the quantity of goods produced at time t , the relative rate of change Q ( t ) Q ′ ( t ) indicates how quickly the production is growing relative to the current production level. A percentage rate of change of 5% would mean the production is growing at 5% of the current level, offering a standardized way to compare growth rates across different scales or industries.
