Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 30

a.) Using Quotient Rule

$
\begin{equation}
\begin{aligned}
S = \frac{dR}{dx} &= \frac{\left(1+4x^{0.4}\right) \frac{d}{dx}\left( 40 + 24x^{0.4}\right) - [ 40 + 24x^{0.4}]\frac{d}{dx} \left(1+4x^{0.4}\right)}{\left(1+4x^{0.4}\right)^2}\\
\\
\frac{dR}{dx} &= \frac{\left(1+4x^{0.4}\right)\left(9.6x^{-0.6}\right) - [40 + 24x^{0.4}] \left(1.6x^{-0.6}\right) }{\left(1+4x^{0.4}\right)^2}\\
\\
\frac{dR}{dx} &= \frac{9.6x^{-0.6} + 38.4x^{-0.2}-65x^{-0.6}-38.4x^{-0.2}}{\left(1+4x^{0.4}\right)^2}\\
\\
\frac{dR}{dx} &= \frac{-54.4x^{-0.6}}{\left(1+4x^{0.4}\right)^2}
\end{aligned}
\end{equation}
$


Therefore, the sensitivity is $\displaystyle s = \frac{dR}{dx} = \frac{-54.4}{x^{0.6}\left(1+4x^{0.4}\right)^2}$

b.)



The graph shows that a small stimulus $x$ produces a large reaction on the body and indicates an extremely high sensitivity which is something we do not expect.