Real problems rarely come with labels. An important skill is to give names to things you don't know. Working in groups is a good way to develop this skill — it's hard to talk about something without giving it a name!

A problem illustrating this technique is included in Group Activity 2 (Acceleration), and discussed in Part III.

Implicit differentiation is easy using differentials — simply take the differential of each term. For instance, if \begin{equation} x^2+y^2=4 \end{equation} then adding the differentials of each term results in \begin{equation} 2x\,dx + 2y\,dy = 0 \end{equation} This equation could be used to find the slope of the tangent line to this circle at any point, by solving for $dy\over dx$. But there is no need to think about which variables are dependent, and which are independent! We find it easiest to think about this process as finding the small change in the entire equation, by determining how each term changes.

Here is an example we often give our students for homework:

The voltage $V$ (in volts) across a circuit is given by Ohm's law: $V=IR$, where $I$ is the current (in amps) flowing through the circuit and $R$ is the resistance (in ohms). If we place two circuits, with resistance $R_1$ and $R_2$, in parallel, then their combined resistance $R$ is given by \begin{equation} {1\over R} = {1\over R_1} + {1\over R_2} \end{equation} Suppose the current is 2 amps and increasing at $10^{-2}$ amp/sec and $R_1$ is 3 ohms and increasing at 0.5 ohm/sec, while $R_2$ is 5 ohms and decreasing at 0.1 ohm/sec. Calculate the rate at which the voltage is changing.