A Derivative Is a Rate, and a Rate Needs Units, a Direction and a Moment

You'll learnto read a derivative's units straight off the difference quotient, and to write the four-part sentence that says what it means — without ever confusing the rate with the amount.

Unit 2 built the machinery and Unit 3 extended its reach. From here a derivative stops being a symbol to push around and starts being a measurement of something real. The number is rarely the hard part. Saying what it means is. Two failures do almost all the damage: a rate reported bare, with no units, no variable and no moment attached — and a statement about M′ read as a statement about M, so a negative rate is taken to mean a negative amount, or a rate is quoted as a total.

8 STEPS · 6 QUICK CHECKS · UNITS COME FROM THE QUOTIENT · WHEN + WHAT + DIRECTION + SIZE · A RATE IS NOT AN AMOUNT · v1

M(t) = 200(0.94)^t mg ⇒ M′(t) = M(t)·ln(0.94) mg per hour

Before you start

What you're looking at

A dose of a drug leaving the body. M(t) is the amount still there, in milligrams, t hours after the dose. The token on the top line is that amount; the arrow pinned to it is M′(t), the rate at which that amount is changing.

The question

Given a number like M′(3) = −10.28, what sentence actually earns the point — and what does that number flatly refuse to tell you?

Watch for

The token stays far to the right of zero the whole time while the arrow points left the whole time. A positive amount and a negative rate, at the same instant.

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