Exercise 7.1: Interpretations of sensitivity
Sensitivity, as explored in Chapter 7, is a metric defined as the ratio of the fractional change in output to the fractional change in input.
a) Following the definition, derive the formula for sensitivity
\[\begin{align} S = \frac{\mathrm{d}\ln y}{\mathrm{d}\ln x}. \end{align}\]
b) Some references describe sensitivity as a relationship between the fold-change in output, \(y/y_0\), and the fold-change in input, \(x/x_0\), where \(x_0\) and \(y_0\) are respectively the input and output values before a change in input value, and \(x\) is the updated input value and \(y\) is the updated output value. Write \(S\) in terms of these respective fold changes.