Derivative Method

Recall equation (1)


Obtain the first derivative dW/dt by differentiating the equation on both sides, i.e.


In relation to the graph W vs. t given by (1), dW/dt at a t-value will give the slope of the tangent at that t-value. What will be the slope of the tangent at the point where a graph has a minimum? Of course, it is zero (if you are not sure about this, review the section on slope.) Therefore, in (2), if we make

one obtains


Now, by solving (3), we can find the t-value


At this juncture, let us pause for a minute. Do we know for sure that this t-value corresponds to the minimum value of W? No, we do not! This t-value may very well correspond to the maximum value for W, because the slope of the tangent at the point where a graph has a maximum is also zero!!

For the case study presented in this lesson, show that the t-value in (4) corresponds to a minimum.

So, the minimum weight will occur 6.5 days after the baby is born and that minimum
weight will be

This means that the baby will start to grow after 6 and a 1/2 days from birth.

Question: Could we have found this result without using the concept "derivative"?

The answer is yes. See how with the derivative-free and algebraic methods.

Trial & Error Approach Derivative-free