Question

What is the relative rate of change?

Answer 1

Hint: For solving this question you should know about the relative rate of change. The relative rate of change of a function \[f\left( x \right)\] is the ratio if it’s derivative to itself.
We can say that the relative rate of change of a function is the ratio of the first derivative of the function with respect to the same function.

Complete step by step answer:
According to our question we have to define what is the relative rate of change.
So, we can write it as the ratio of the first derivative of any function with the same function it means,
\[R\left[ f\left( x \right) \right]=\dfrac{f'\left( x \right)}{f\left( x \right)}\]
For understanding it clearly we can take an example.
If \[f\left( x \right)={{x}^{2}}\]
then \[f'\left( x \right)=\dfrac{d}{dx}f\left( x \right)\]
So, the first derivative of the function is:
\[f'\left( x \right)=\dfrac{d}{dx}\left( {{x}^{2}} \right)=2x\]
So, the rate of the change \[=\dfrac{f'\left( x \right)}{f\left( x \right)}\]
Since, \[f\left( x \right)={{x}^{2}}\]
 \[f'\left( x \right)=2x\]
So, the relative rate of change \[=\dfrac{2x}{{{x}^{2}}}=\dfrac{2}{x}\]
It can also be used in the form:
\[\dfrac{f'\left( x \right)}{f\left( x \right)}={{\left[ \ln f\left( x \right) \right]}^{\prime }}\]
If \[f\left( x \right)={{x}^{2}}\]
By solving this,
\[\begin{align}
  & {{\left[ \ln f\left( x \right) \right]}^{\prime }}={{\left[ \ln {{x}^{2}} \right]}^{\prime }} \\
 & ={{\left[ 2\ln x \right]}^{\prime }} \\
 & =\dfrac{2}{x} \\
\end{align}\]
Thus, we can calculate the relative rate of change of a function with the help of the first derivative and the function.

Note: The relative rate of change is the ratio of first derivative of the function and the function. So, these both are independent but the relative rate of a function is dependent on both of this. So, it decides how many functions will be changed with respect to its own derivative. In this the first derivative is always considered as a numerator and the main function will always be considered as a denominator. And for the logarithmic functions this is also the same as other functions.