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.

Answer and Explanation:

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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}...

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Quotient Rule: Formula & Examples

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Chapter 1 / Lesson 5
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