Reading a derivative as how fast one quantity answers another

0:00Everything you have learned about differentiating was about how to compute one thing. This is about how to read it. A derivative is a rate. It is a sentence about two quantities: how fast the first one answers a change in the second. Nothing new is being defined here. The object is the same object. What changes is what you say out loud when you look at it. And that turns out to be harder than the computing, because the computing has one answer and the reading has three parts: which quantity, against which other quantity, in what unit.

0:41Get the first two right and the third comes out on its own. Get the second one wrong and you can do every line of the arithmetic perfectly and still write down an answer that is not the answer to any question anybody asked. That is not a hypothetical. It is what happens in the very first worked example of this topic, and we are going to take it apart carefully, because the mistake in it is the mistake everybody makes.

1:14One short warning before we start, and then we will not mention it again. The opening paragraph of this part of the course promises several things. Some of them arrive. You will read derivatives as rates, which is this video. You will find where a quantity turns round, and use that to find its largest and smallest values. You will learn to say where a quantity is rising and where it is falling.

1:42Two of the promises do not arrive at all. There is nothing on the two standard lines you can draw at a point of a curve, and nothing on using a derivative to approximate a value. They are announced and then never taught. If you have been waiting for them, stop waiting. They are good topics and they are somewhere else. Start with the one rate you already read without translating.

2:11Distance against time. If a distance covered is three times the square of the time taken, plus five times the time, then the rate at which the distance answers the time is six times the time, plus five. At a time of two, that rate is seventeen. Seventeen what? Seventeen centimetres per second. You did not have to think about that unit. It came free, because the two quantities were named in the question and the word per did the rest.

2:45That is the whole skill, and you already have it. The rest of this video is about what happens when the second quantity is not time, which is when the free unit stops being free. Now widen it. Replace distance by any quantity at all, and replace time by any other quantity, provided one thing: a rule tying the two together, so that fixing one fixes the other. The rate is then the derivative of the first with respect to the second.

3:19With respect to. Those three words are the whole of the new idea, and they are the words most often skipped when the sentence is read aloud. Say them. They name the second quantity, and the second quantity is what the answer will be measured against. An area with respect to a radius. A volume with respect to a radius. A total cost with respect to the number of units made.

3:48Three different second quantities, three different units in the answer, and not a second of time among them. Before the first example, one distinction that costs marks every year. A derivative is a function. Its value at one input is a number. They are two different objects and they get two different pieces of notation. Written as a ratio with nothing attached, it means the function: the rate at every input at once.

4:21Written with a vertical bar and a small subscript naming the input, it means one number: the rate there and nowhere else. The primed forms do the same job, one with the letter in brackets and one with a particular value in brackets. Here is why it matters. In this topic there are twelve rates measured, spread across seven quantities, and no quantity takes the same rate at two of its own inputs.

4:50Not one. So if a question names an input and you hand back the function, you have not given a slightly incomplete answer. You have named none of the values it can take. Now the first worked example, done twice. Once for the number, which is easy, and once for the units, which is where the whole topic lives. The number first. A circle's area is the circle constant times the square of the radius.

5:22Differentiate with respect to the radius and you get twice the constant times the radius. At a radius of five, that is ten times the circle constant. Nothing here is quoted. The checker behind this video never writes the circle constant down; it finds a quarter turn as the first input at which the cosine vanishes, by halving, and doubles it. And it never quotes a derivative either. It takes the difference quotient, shrinks the neighbourhood from both sides of the point until it settles, and only then compares what it settled on with what the algebra claimed.

6:05The two agree. Ten times the circle constant it is. And the same circle, asked at radii of three, four and six, gives six, eight and twelve times the constant. Now the same example again, for the units, and this is where it goes wrong. The question asks how fast the area is changing per second. The working differentiates with respect to the radius. And the answer that gets written down carries square centimetres per second.

6:38Look at those three lines together. The question names a second. The working takes no rate against time anywhere. And the answer carries a unit that the working never produced. So what unit did the working produce? An area, over a length. Square centimetres over centimetres. Which is centimetres. The rate at which a circle's area answers its radius is a length. That reads oddly the first time and it is exactly right: the answer is ten times the circle constant centimetres.

7:16Not per second. There is no second in the problem. And the same circle gets asked again straight afterwards as an exercise, with no mention of a second at all, which is the question the worked example meant to ask. Now, it would be easy to say all that and expect you to take it on trust. So it was measured instead. The checker behind this video carries a unit as a real object through every line: a quantity is a value together with the powers of the four base units it is measured in.

7:55A difference keeps the unit both its sides carry, and refuses outright to subtract two unlike things. A rate subtracts the two unit vectors. That one line is the whole arithmetic of units, and the unit of every answer is then never stated anywhere in the file. It is whatever the difference quotient's own arithmetic produces. Then four readings of the question what unit does this rate carry were scored against that produced unit, by identical machinery, at all twelve rates in the topic.

8:32Reading it off the two quantities is right at twelve of twelve. The reflex that a rate is always per second is right at none of them. Zero of twelve. Leaving the first quantity's unit unchanged is right at five of twelve. And multiplying the two units instead of dividing them is also right at five of twelve. Those two are not right at random. They are right at exactly the five money rates and nowhere else, and we will come back to why.

9:07There is a second way to ask the same question, and it is worth having, because it does not mention units at all. A rate is not a label. It is a prediction. It says: move the second quantity by a small amount, and the first quantity will move by the rate times that amount. That is testable, and it was tested. At all twelve rates the change in the first quantity really is the rate times the increment, up to a leftover that shrinks at least fiftyfold when the increment shrinks tenfold.

9:45Second order, in other words. The same test put to those twelve answers with a thousandth added fails at every single one, because their leftover is first order and shrinks only tenfold. Now run the units through the same prediction. Whatever unit the rate carries, multiplying it by the increment's unit has to give back the unit of the thing that changed. That is a product, where the reading was a quotient, so it is a different door.

10:17And the five readings score twelve, seven, nothing, five and five through it, exactly as they did before. Take the answer that got written down through it: square centimetres per second, times an increment of length, is a volume per second. The change in an area is not a volume per second. The produced unit, centimetres, times an increment of length, is square centimetres, which is exactly what changed. Here is the part that says something about how this topic is learned rather than about the mathematics.

10:57Of the twelve rates, at five of them at least one of the three wrong readings gives the right unit anyway. Only seven rule out all three at once. And the five are precisely the money ones. So if you practise on the examples that feel easiest, you can get every unit right for the wrong reason, and never find out. That is a general shape and it has turned up in every topic of this course so far.

11:26An example chosen because it is easy to compute is very often an example that cannot tell a rule from its rivals. The seven rates that do rule everything out are the geometric ones, the areas and the volumes against lengths, which is why they are worth doing slowly even though the arithmetic is trivial. One boxed sentence sits in the middle of this topic and it is the only interpretive remark in it.

11:59A rate is positive when the two quantities move the same way, and negative when they move opposite ways. That is worth thirty seconds on its own, because it is the sentence that will later be promoted into a theorem. It was measured, not repeated. Eight points were set up: a total cost rising with output, a height falling from rest, a square of a distance on the left of its lowest point and on the right of it, a reciprocal, a quantity ten below its input, an input turned round, and the lowest point itself.

12:38At every one of the eight, the sign of the measured rate agrees with which way the first quantity moves. Including the point where the rate is nought and the quantity moves neither way. And that second reading is taken from the values on the two sides of the point, with no derivative in it anywhere, so it really is a second opinion. The misreading was scored too. A negative rate does not mean the quantity is negative.

13:08Of the four points here whose rate falls, the quantity itself is positive at three and negative at one. So the reading is right once and wrong three times. And the quantity ten below its input is negative while rising, which catches it from the other side. Now change the second quantity to something with no length and no time in it at all. Money against output. A total cost is a rule in the number of units made.

13:41The rate at which that total answers one more unit of output is called the marginal cost. A total revenue is a rule in the number of units sold, and the rate at which it answers one more unit sold is the marginal revenue. Notice what the definitions say. They say rate. They do not say difference. Marginal cost is a derivative, evaluated at the output level you are currently at.

14:11It is an instantaneous rate, in exactly the sense that a speed is an instantaneous rate. Take the first one. A total cost of five thousandths of the cube, less two hundredths of the square, plus thirty times the number, plus five thousand. Differentiate: fifteen thousandths of the square, less four hundredths of the number, plus thirty. At three units, that is thirty and fifteen thousandths. And the second: twenty-one thousandths of the square, less six thousandths of the number, plus fifteen, which at seventeen units is twenty and nine hundred and sixty-seven thousandths.

14:56And here is the reading almost everybody arrives with. Marginal cost is the cost of the next unit. It sounds right. It is even how the word marginal is used outside mathematics. It is a different number. So rather than telling you that, four readings were put to the same five money rates by identical machinery: the derivative taken from the definition, the cost of the next unit, the cost of the unit just made, and the average of those two.

15:31They score five, nothing, nothing, and three. The derivative is right at all five, which is what it means for the definition to be the definition. The cost of the next unit is right at none of them. The cost of the unit just made is right at none of them. And the two whole-unit readings are wrong in opposite directions at every one of the five: the next unit always above the rate, the unit just made always below.

16:02So neither of them is a small error in the other's direction. They straddle the answer without ever landing on it. The fourth reading is the interesting one. The average of the two neighbouring differences is right at three of the five. Three of five is a strange score. It is not almost right and it is not wrong. So which three? Exactly the three revenue rates. And that is not a coincidence, it is a fact about squares.

16:36If a total is a square in the number sold, then the average of the difference forward and the difference back is the derivative exactly, with nothing left over. Both revenues in this topic are squares, so the average is not approximately right there; it is right. If a total is a cube, it is not. The average exceeds the derivative by the cube's own leading coefficient: five thousandths on the first cost, seven thousandths on the second.

17:10Both of those were asserted as exact differences rather than reported as small numbers. So the reading is right for a reason that has nothing to do with marginal cost, and wrong the moment the total stops being a square. And now the part that made this worth a scene of its own. These are money answers, and money answers get rounded to two decimal places. Round them. Scored on the exact numbers, the average of the two neighbouring differences is right at three of the five money rates.

17:47Scored on the answers the way anybody actually writes them, rounded to two places, it is right at all five. All five. The rounding hides the one error that matters. There is nothing a reader is ever shown, anywhere in this topic, that can tell that reading from the derivative. And that is not rounding making everything agree: the two whole-unit readings are still wrong at all five after rounding, and the same rounding put to the answers a thousandth of themselves out separates them at all five as well.

18:26Two places is enough to see a thousandth. It is not enough to see the reading that is actually wrong here. There is a second thing hiding in that rounding. The first money answer, thirty and fifteen thousandths, sits exactly halfway between thirty and one hundredth and thirty and two hundredths. Two hundred times it is an odd whole number. So the rounding rule, and not the arithmetic, decides its last digit.

18:56Round half away from nought and you get thirty and two hundredths. Cut the digits off and you get thirty and one hundredth. If you have ever wondered why a rounding convention is worth stating, that is why. Back to the units for a moment, because the money examples answer the question we left open. A total revenue of three times the square plus thirty-six times the number plus five has a rate of six times the number plus thirty-six.

19:29At five units sold that is sixty-six. Sixty-six what? Just sixty-six, in currency. No compound unit at all. Compare that with the circle, whose rate came out as a length. The difference is what the second quantity is. Count the units sold as a plain number and the rate is currency divided by nothing, which is currency. Give a unit of output a unit of its own and the same rate is currency per unit.

20:01Both are defensible, and the two conventions were run through the same difference quotient here: they give currency per unit at the five money rates, and they agree exactly at the seven geometric rates. So the choice changes the answer at the money rates and nowhere else. The first convention is the one in ordinary use, which is why marginal answers come out as plain figures of currency. That is also why leaving the first quantity's unit unchanged scored five out of twelve earlier.

20:38It is right exactly where dividing by the second quantity divides by nothing. The exercise closes with two multiple-choice items, and each one is checking something different. The first asks for the rate at which a circle's area changes with its radius, at a radius of six. Twelve times the circle constant. That one is checking whether you can do the topic at all. The second is the revenue we just did, at fifteen units instead of five.

21:14One hundred and twenty-six. That one is checking something else entirely. It is the worked example with a single number changed, so if you understood the example rather than copying it, you can answer it without writing anything down. If you cannot, that is useful information, and it is information about the example and not about the question. Two more from the same exercise, both rates against a length and not against time: a sphere's volume against its radius, at a radius of ten, is four hundred times the circle constant.

21:54And a sphere whose diameter is one and a half times one more than twice a length has a volume rate of twenty-seven eighths of the constant times the square of that same bracket, which is two hundred and forty-three eighths at a length of one, and six hundred and seventy-five eighths at two. Every one of those was settled by its own difference quotient before it was written down.

22:23One thing this topic deliberately does not do, so that you know it is missing rather than assuming you missed it. Every rate here is taken against a length or against a count. Not one of the twelve is taken against time. That was measured, not asserted: a second of time appears in no quantity and in no second quantity anywhere in the topic. But the sentence we started from was about time, and rates against time are where this all goes next: a radius growing while you watch, and an area that answers it through a chain of two rates rather than one.

23:04That is the next topic, and everything in it rests on being able to say which quantity you are taking a rate against. Which is what this one was for. Three things to take away. First, a derivative is a rate, and reading it aloud means naming two quantities and not one. With respect to what? If you cannot answer that, you have not read it yet. Second, the unit is not decoration and it is not looked up.

23:36It is produced by dividing the first quantity's unit by the second's, and it will tell you when you have taken the rate against the wrong thing, which is exactly what happened in the first example of this topic. The number there is right and the unit is not, and the two do not disagree by accident. Third, marginal means a rate at an instant, not the cost of one more unit.

24:03Those are different numbers, and on the cost examples here rounding to two decimal places makes one of the wrong readings indistinguishable from the right one, which is a reason to know the definition rather than to pattern-match the answers. None of the numbers in this video were copied from anywhere. A checker written before the narration measures every rate as the limit of a difference quotient from both sides, finds the circle constant by halving rather than quoting it, and carries a unit through every line as a real object.

24:43It records forty claims, none wrong, seventeen liveness controls, and four of four deliberately planted faults caught. A separate probe plants one hundred and two deliberate defects in it, one at a time, and reruns it against each; all one hundred and two are caught. Alongside them run fifteen rewrites that change nothing this topic can see, and none of those is wrongly caught.