Question

allometric equations describe the scaling relationship between two measurements, such as skull length versus body length. in vertebrates, we typically find that skull length ∝ body length^a for 0 < a < 1. express the growth rate of the skull length in terms of the growth rate of the body length. (type an expression using s and b as the variables.)
\\( \frac { d s } { d t } = \frac { \frac { d s } { d b } } { \frac { d t } { d t } } \\)

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). In this case, let \(S\) be a function of \(B\) (\(S = S(B)\)) and \(B\) be a function of \(t\) (\(B=B(t)\)). Then, by the chain rule, \(\frac{dS}{dt}=\frac{dS}{dB}\cdot\frac{dB}{dt}\).

Step2: Solve for \(\frac{dB}{dt}\)

We want to express \(\frac{dB}{dt}\) in terms of \(\frac{dS}{dt}\) and \(\frac{dS}{dB}\). Rearranging the equation \(\frac{dS}{dt}=\frac{dS}{dB}\cdot\frac{dB}{dt}\) for \(\frac{dB}{dt}\), we get \(\frac{dB}{dt}=\frac{\frac{dS}{dt}}{\frac{dS}{dB}}\)

Answer:

\(\frac{dB}{dt}=\frac{\frac{dS}{dt}}{\frac{dS}{dB}}\)