The rate of change at a point, defined as a limit of average rates

0:00How fast is something moving right now? Not over the last second, not over the last hundredth of a second - right now, at this instant. Average speed is easy: pick two moments, take the distance between them, divide by the time between them. But an instant has no two moments in it. The distance you would divide is nothing, and the time you would divide it by is nothing as well, and nothing divided by nothing is not a number.

0:32So the question as asked has no answer. What follows is the repair, and it is one of the most useful ideas anyone has ever had. You do not measure at the instant. You measure near it, over a gap you then shrink, and you watch where the readings are heading. Here is the machinery. Take a function f and a place a in its domain. Step away from a by an amount h, and compare the two heights: f of a plus h, minus f of a.

1:05Divide that by h, the width of the step you took. That ratio is the average rate of change across the step, and it is called the difference quotient. Now shrink h toward nothing and ask what the quotient settles on. If it settles on something, that something is the derivative of f at a. Watch the last clause, because the whole rest of this video is inside it. Provided the limit exists.

1:35Not every function has one, not everywhere, and that is not a technicality tucked into the small print - it is a fact you can measure. Before anything else, look at what the quotient does at h equals nothing. The top becomes f of a, minus f of a - nothing. The bottom is h, which is also nothing. So substitution is not merely awkward here, it is unavailable. You might think that is a feature of the one example you tried.

2:09It is not, and here is the measurement. Fifteen functions were put to forty-three places each, and between them those allow six hundred and forty-one readings. The number of readings whose top fails to shrink to nothing is zero. The number that substitution can deliver is also zero, and when it is asked it says the same thing every time: there is nothing to divide by. Every single derivative in this video has to be got some other way.

2:43Start with the friendliest possible case. f of x is three x, and we want its derivative at two. Step away by h: three times two plus h, minus three times two, is three h. Divide by h and the quotient is three. And notice what just happened - the h vanished completely. The quotient does not depend on how big a step you took. Across the twelve step sizes used throughout this video, from a quarter down to one part in eight thousand one hundred and ninety-two, the number of them that read anything other than three is zero.

3:24So the readings do not approach three, they are three, and the derivative at two is three. That is the answer, and it came out of the definition and nothing else - no rule, no table, no pattern noticed from a list. Most cases are not that kind. Take f of x equals x squared. The top is x plus h, all squared, minus x squared, which is two x h plus h squared.

3:54Every term carries an h, which is exactly what it means for the top to shrink to nothing - and that is what licenses the next step. Take one factor of h out of the top and cancel it against the bottom. What is left is two x plus h, and now h can go to nothing without any trouble at all: the derivative is two x. That cancelling is the whole method, and it is worth seeing what it costs.

4:24Across the eleven functions here built from powers, the tops of their quotients have degrees running zero, one, one, one, one, one, two, two, two, two and three. After the factor comes out, those become zero, zero, zero, zero, zero, zero, one, one, one, one and two. One power off each, every time. Cancelling is an algebraic move, and it is fair to ask whether it is giving the right answer or merely a tidy one.

4:59So here is a second route that never cancels anything. Take the quotient at a run of ever-smaller steps, from both sides, and use exact elimination to squeeze out the leftover terms - arithmetic only, no factorising, no algebra on the expression at all. Put the two routes side by side over the functions built from powers. Four hundred and seventy-three readings, and the number where the two disagree is zero.

5:32Now, a zero like that is worth nothing until you know the comparison could have reported something else. So halve the second route's answer on its way out and score it again. Now four hundred and twenty-seven readings disagree, and the forty-six that still agree are exactly the forty-six where the derivative is nothing - because half of nothing is still nothing. The comparison works. The agreement was real. Then there are the four functions here that are one thing over another - one over x, and its relatives.

6:10Between them they allow one hundred and sixty-eight readings. Cancelling handles every one of them. The run of shrinking steps handles none of them. Not one, out of a hundred and sixty-eight - and it does not go quietly, it says the readings do not settle on any one value. That is worth sitting with, because it inverts the usual picture. The numerical route feels like the honest one and the algebra like the shortcut.

6:41Here it is the other way round: for these functions the quotient is not a polynomial in h at all, so no finite amount of squeezing ever clears it, and taking the factor out is not a convenience. It is the only exact route there is. Every derivative in this video has an answer that came from somewhere else - written down first, then checked. Scored against what the definition gives, over all six hundred and forty-one readings, the number that differ is zero.

7:17Again the same question: could that zero have been anything else? Halve the quotient before comparing, and five hundred and ninety-five of them differ. Better still: give each function the next function's answer instead of its own, and out of six hundred and forty-one readings the number that still happen to agree is two. Shifting the answers by one position destroys six hundred and thirty-nine agreements out of six hundred and forty-one.

7:49So the answers are not being generated from the definition and then compared with themselves. They are separate, and they match. With that settled, here are the cases the definition delivers. Two x squared plus three x minus five: at minus one the derivative is minus one, and at nothing it is three. Add three times the first to the second and you get nothing exactly - a relation you can now check rather than take on faith.

8:22A constant function, f of x equals three: at nothing the derivative is nothing, and at three it is nothing again. That is not because there is nothing to differentiate - the quotient gets computed like every other, and the top is three minus three. Ten x gives ten. x squared minus two, at ten, gives twenty. x itself, at one, gives one. Ninety-nine x, at a hundred, gives ninety-nine.

8:53And x cubed minus twenty-seven, at two, gives twelve. Now the picture, because there is one and it is the reason any of this matters. Mark P on the curve at a, and Q further along at a plus h. Drop a horizontal from P and a vertical from Q and they meet at a corner. The vertical side of that right triangle is f of a plus h minus f of a, and the horizontal side is h.

9:25So the ratio of the two sides is the difference quotient itself - the slope of the straight line through P and Q, the chord. Through P there is also the tangent, and the derivative is the slope of that. Careful there: the derivative is the slope, a number - not the tangent line. As h shrinks, Q slides down toward P and the chord tips over onto the tangent, which is the picture of the limit.

9:55On a curve that bends upward, the chord and the tangent are not interchangeable, and which is steeper is not a matter of taste. Take x squared, at forty-three places, with seven step sizes at each: three hundred and one chords. Stepping forward, the number steeper than the tangent is three hundred and one - all of them. Stepping backward, the number steeper is zero - none of them. And you can watch the gap close: at a step of a half the chord beats the tangent by a half, and at a step of one over a hundred and twenty-eight it beats it by one over a hundred and twenty-eight.

10:38Now do the same on a straight line. Three hundred and one chords again, and the number steeper than the tangent is zero going forward and zero going back - because every single one of the three hundred and one is exactly equal to it. The bending is what makes the chord differ from the tangent at all. So far the derivative has been a number attached to a place.

11:07Leave the place as a variable instead and the same limit defines a new function, f prime of x, whose value at each place is the derivative there. This is worth separating carefully, because they are written almost the same and they are not the same kind of thing. f prime of a is a number. f prime of x is a function. You will meet it written several ways - f prime of x, d by d x of f of x, d y by d x when y is f of x, and a capital D in front of f of x.

11:46All the same object. And it is a genuine fact, not a restatement, that reading the derivative function at a place gives the derivative at that place: over all six hundred and forty-one readings, the number where those two disagree is zero. Now back to the clause: provided the limit exists. Here is a function where it does not. Take the absolute value of x - the V shape, with its corner sitting at nothing.

12:19Coming in from the left, every reading of the quotient is minus one. Coming in from the right, every reading is plus one. Neither side is confused, neither side wanders, and they settle on different numbers - so there is no single value for the quotient to approach, and the derivative at nothing simply does not exist. Everywhere else it is perfectly well behaved. Out of the forty-three places swept, the number where this function has a derivative is forty-two.

12:53Being able to draw a function, even with one clean unbroken stroke, is not the same as being able to differentiate it. A derivative can go missing in more than one way, and lumping them together loses the plot. Take a function that jumps: nothing below one, and one from there on. At the jump, the top of the quotient does not shrink at all - coming in from below it is minus one at the widest step and still minus one at the narrowest.

13:28So the quotient runs away: at the widest step it reads four, and at the narrowest it reads eight thousand one hundred and ninety-two, and it will beat any number you name. Three different failures, three different things to say. The V shape: the two sides settle on different values. The jump: the readings do not settle on any one value. And one over x at nothing: that input is not in the domain.

13:57Only the last of those is the function's fault before you even start. Which raises a question people get wrong in a specific way. Does the derivative function have the same domain as the function? The example usually offered is one over x, which is undefined at nothing and whose derivative, minus one over x squared, is undefined at nothing too. And measured, that is exactly what happens: forty-two places where the function can be read, forty-two where its derivative can be, and the two sets are identical.

14:39But that is an exclusion inherited, not a new one. The function was already missing there. For a case where the derivative's domain is genuinely the smaller, go back to the V shape: forty-three places where the function can be read, forty-two where its derivative can be. One place lost, and lost by the differentiating, not by the function. That is the real content of the warning. One last case, and it breaks the method.

15:12The derivative of sine x at nothing. The quotient is sine of h, over h, and the top does shrink to nothing exactly as before - at a tenth of the way in it sits at nought point nine nine eight three three, cut to six places, and it keeps falling. But there is no factor of h to take out. A sine is not built from powers of x, so nothing cancels, and the one move that has carried every case so far is unavailable.

15:44What is left is to trap it. At a tenth of the way in the quotient sits below one by nought point nought nought one six six five eight three three. At a hundredth, by nought point nought nought nought nought one six six six six. At a thousandth, by nought point nought nought nought nought nought nought one six six. Offer it six bounds, each a tenth of the one before, and the number it never gets past is zero - it clears them after one step, three, four, six, eight and nine, and stays clear.

16:19Put the same six bounds around the number two instead and it clears none of them. The derivative is one. Step back and look at what the definition has done. It turned a question with no answer - speed at an instant - into a question with a procedure: form the quotient, watch the top shrink, get rid of the h one way or another, and read what is left.

16:46Every derivative in this video came out of that and nothing else. No rule for a power, no rule for a sum, no table. Those are coming, and they are enormously convenient, and every one of them will be a theorem proved from this. The clause matters too. Provided the limit exists is where the V shape and the jump live, and keeping it in sight is what stops the derivative from becoming a formula you apply without looking.

17:18Form the quotient. Shrink the step. See where the readings are heading. That is all of it.