# The proportion w(t) of offspring hatching per unit of time is given as a function of the...

## Question:

The proportion w(t) of offspring hatching per unit of time is given as a function of the probability f(t) of hatching if time t is spent brooding and the cost C associated with the time spent searching for other mates : {eq}w(t)= \frac {f(t)}{(C+t)}. {/eq} Find the derivative of w(t).

## Differentiation:

To differentiate the function given, we need to make use of the quotient rule of differentiation. If u and v are both functions of x, the derivative is:

\begin{align} &\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{u}{v} \right )\\ &=\frac{vu'-uv'}{v^2} \end{align}

If we look at the function given, we have:

\begin{align} u=f(t)\\ v=(C+t) \end{align}

Both are functions of t.

To find the derivative of the function w(t), we use the quotient rule of differentiation.

\begin{align} w'(t)&=\frac{\mathrm{d} }{\mathrm{d} t} \left (\frac {f(t)}{(C+t)} \right )\\ &=\frac{(C+t)*\frac{\mathrm{d} }{\mathrm{d} t}f(t)-f(t)*\frac{\mathrm{d} }{\mathrm{d} t}(C+t)}{(C+t)^2}\\ &=\frac{(C+t)f'(t)-f(t)}{(C+t)^2} \end{align}