Rules for differentiating a sum, a product and a quotient

0:00There are four rules for combining limits, and four rules for combining derivatives, and they are laid out to look like each other. The limit of a sum is the sum of the limits. The limit of a product is the product of the limits. Four tidy statements, each one saying: do it to the pieces, then combine. The derivative rules copy the first two exactly. And then the copy breaks.

0:28The derivative of a product is not the product of the derivatives, and the derivative of a quotient is not the quotient of them either. This is not a quirk of notation. There is a reason, it is short, and it is the whole of this video. So: two rules that behave, two that do not, and the difference between them measured rather than asserted. Start with the two that behave.

0:57Here are nine functions. Constants, straight lines, squares, a cube, one over x, a ratio. Pair each with each and you get eighty-one ordered pairs, and each pair is read at thirty-seven places. Now build the sum of each pair, take its derivative straight from the definition, and set that beside the sum of the two separate derivatives. Two thousand nine hundred and sixty-three readings agree. Zero disagree. Do the same for the difference and the numbers are identical: two thousand nine hundred and sixty-three agreements, and nothing that differs.

1:38So for sums and differences the copy is exact. Differentiate the pieces, add or subtract, done. Now the product. And before the rule, the disproof - because the wrong answer is the one almost everyone reaches for first. If the derivative of a sum is the sum of the derivatives, surely the derivative of a product is the product of the derivatives. Take both functions to be x. Their product is x squared.

2:11At minus one, the derivative of x squared is minus two. At a half it is one. At two it is four. Now multiply the two separate derivatives. The derivative of x is one. One times one is one. One, one, one. Against minus two, one, four. Across all thirty-seven places the two agree at exactly one of them. That one place is x equals a half, where the true answer happens to be one as well.

2:43One coincidence out of thirty-seven. The rule is not approximately right. It is wrong. So why does the product not split? Because a derivative is not a limit of a product. It is a limit of a difference. Move x along by a step. The product changes from the old first times the old second, to the new first times the new second. That difference has two things changing at once, and there is no way to read it as one change.

3:17But there is a trick, and it is pure algebra with no calculus in it. Add a middle quantity and take the same middle quantity straight back off again. Nothing has changed, because you added zero. And now the difference falls apart into two pieces: the change in the first, times the new second, plus the old first, times the change in the second. Checked over every pair, at every place, at two different step sizes: five thousand eight hundred and ninety-two readings.

3:53The two pieces come to the whole difference in every single one. Zero failures. Two changes. Two pieces. That is where the two terms come from, and they arrive before any limit is taken. There is a second way to do it, and it is worth seeing. Add and take away a different middle quantity, and the difference falls apart into a different pair of pieces. Take both functions to be x again, at the place one, with a step of a half.

4:26Broken the first way, the first piece is three quarters. Broken the second way, the first piece is a half. Different pieces. But both pairs total five quarters, and five quarters is exactly what the product actually changed by. Scored over all eighty-one pairs and every place, the second break lands where the first one did, to the reading. Which is why the product rule can be written with either factor differentiated first. Both are the same statement.

5:00And it tells you what the wrong rule was doing: multiplying the two changes together instead of pairing each change with the other function. That accounts for the whole difference in one hundred and fifty-one readings and fails in five thousand seven hundred and forty-one. Divide those two pieces by the step, let the step shrink, and each piece settles. The first differentiated, times the second. Plus the first, times the second differentiated.

5:35That is the product rule. Two terms, and each term differentiates exactly one of the factors. Now score it the same way the sum rule was scored: build the product, take its derivative from the definition, and compare against what the rule predicts. Two thousand nine hundred and sixty-three agreements. Zero disagreements. The same eighty-one pairs, the same thirty-seven places, the same measuring line. And the rule almost everyone reaches for first, put through that identical line: one hundred and sixty-four agreements, and two thousand seven hundred and ninety-nine disagreements.

6:19One rule gets everything. The other gets a small fraction of it, by accident. Those one hundred and sixty-four are worth a moment, because a wrong rule that is sometimes right is more dangerous than one that is always wrong. The easy explanation is that one of the two derivatives was zero, which would make both sides collapse to nothing. That covers fifty-five of them. It leaves one hundred and nine agreements with no excuse at all - places where two genuinely different calculations happened to land on the same number.

7:01So you cannot test a rule by trying it once and finding it works. If you had tested the wrong product rule on x squared at a half, it would have passed. One reading is not evidence. Two thousand nine hundred and sixty-three readings, with nothing that differs, is. The rule gets written down in a compressed way, and it is worth being fluent in it. Call the first function u and the second v. Write a dash for a derivative.

7:33Then the product rule is: u v, dashed, equals u dash v, plus u v dash. A handful of symbols, and it says exactly what the long version said. It carries Leibnitz's name, after one of the people who built this subject. Read it out loud as you write it: first differentiated times second, plus first times second differentiated. The order of those two terms does not matter, because they are added.

8:04Hold on to that sentence, because the next rule takes it away. The quotient rule. On top: the lower function times the derivative of the upper, minus the upper times the derivative of the lower. Underneath: the lower function, squared. Three things to notice, and all three are places people slip. The top is a difference, not a sum. The order matters. The bottom is the lower function squared - the function itself, not its derivative.

8:39And the whole thing needs the lower function not to be zero, which is a condition the product rule never had. Scored across the same eighty-one pairs and thirty-seven places: two thousand eight hundred and ninety-six agreements, zero disagreements. And one hundred and one readings refused, which is the condition doing its work rather than being ignored. Now take that sentence - the order of the two terms does not matter - and carry it straight into this rule.

9:14Write the top the other way round: the upper times the derivative of the lower, minus the lower times the derivative of the upper. Through the same measuring line: four hundred and thirteen agreements, and two thousand four hundred and eighty-three disagreements. It is a minus sign, and it turns the answer over. The second slip is the denominator. Square the derivative of the lower function instead of the lower function itself.

9:46Four hundred and forty agreements, two thousand one hundred and seventeen disagreements - and four hundred and forty refusals, because that denominator is zero in places the real one never is. Two small changes, two broken rules, both checked the same way as the real one. Here is the check that tells you whether you have actually understood this, and you can run it on yourself. Take one function and take it apart two different ways. The answers have to agree.

10:21Ten x. First route: ten x is x added to itself ten times. Apply the sum rule ten times. The derivative of x is one, so you get ten copies of one, which is ten. Second route: ten x is the constant ten multiplied by x. Now it is a product, so use the product rule. The derivative of a constant is zero, and the derivative of x is one.

10:50Zero times x, plus ten times one. Zero plus ten. Ten. Checked at every one of the thirty-seven places, the two routes disagree at none of them. Same function, two decompositions, one answer. If yours ever differ, you have made an arithmetic slip - not found a new function. Run it again on x squared. The product rule route: x squared is x times x. One times x, plus x times one. That is two x.

11:25The other route is the definition itself - form the quotient, let the step go to nothing. At minus two, both give minus four. At zero, both give zero. At three, both give six. Across the whole sweep of places, the number of disagreements is zero. That matters more than it looks. The rules were never proved here - they were stated. Every time a rule and the definition land on the same number, the rule has survived a test it could have failed.

11:59A worked ratio: x plus one, over x. The upper function differentiates to one. So does the lower. Both are just one. So the top of the quotient rule is x times one, minus x plus one times one. The x and the x cancel and you are left with minus one. The entire numerator collapses to a single negative digit. Underneath, x squared. So the derivative is minus one over x squared.

12:32At two that is minus a quarter. At minus three it is minus a ninth. Always negative, because minus one over a square cannot be anything else. And across the thirty-seven places it refuses at exactly one of them, giving the reason: that input is not in the domain. That hole is not something differentiating created. The ratio never had a value at zero. The derivative inherited the exclusion, and the rule carried it through rather than quietly filling it in.

13:10Now trigonometry, and here I have to be careful about something. It would be very easy to check the product rule on sines by using the known derivatives of sine and cosine - but those are what the rules are supposed to be delivering. A check that assumes its own answer proves nothing. So nothing in what follows knows what the derivative of a sine is. Instead: build the difference quotient out of the sine itself, walk the step inward through twelve sizes, from a quarter down to one part in eight thousand one hundred and ninety-two, and ask a different question.

13:51Not what does it equal. Rather: how far in do I have to go before it sits inside a given bound of what the rule predicts, and stays inside. Six bounds, each tighter than the last. Start with the sine on its own, against the cosine, worked out by a completely separate calculation. All six bounds cleared, at steps one, two, three, four, five and six. And to prove that test can fail: the same quotient against the cosine with its sign flipped clears none of the six.

14:28Sine squared. Written as sine times sine, so the product rule applies. Both factors are the same function, so both terms of the rule come out the same: a sine times a cosine, twice over. At a half, that prediction is nought point eight four one four seven zero nine eight four. Squeeze the real quotient onto it and all six bounds clear, at steps one, three, four, five, six and eight.

14:59Slightly deeper than the plain sine needed, which is what you would expect from a more complicated shape. The rule predicted a number it had no way of knowing, and the function walked onto it. And now the trap, which is the single most common confusion in this corner of the subject. That prediction - nought point eight four one four seven - is exactly the value of the sine of twice the angle at that place.

15:29Worked out by a separate calculation, to nine decimal places, the two match. So the derivative of sine squared is the sine of twice the angle. A genuinely pretty result. And it is precisely why people then treat the two as interchangeable. They are not the same function. At a half, sine squared is nought point two two nine eight four eight. The sine of twice the angle is nought point eight four one four seven zero.

16:02And their derivatives are not the same either. Rewrite the sine of twice the angle as twice a sine times a cosine, apply the product rule, and the prediction is one point zero eight zero six zero four. Squeezed, it clears all six bounds, at steps four through nine. Now take the wrong step deliberately: predict the derivative of sine squared using the derivative of the sine of twice the angle.

16:33Zero of the six bounds cleared. Not close. Never gets there. One more, and it puts every piece together: cosine over sine. Quotient rule. The top is sine times the derivative of cosine, minus cosine times the derivative of sine. That comes to minus a sine squared, minus a cosine squared - and a squared sine plus a squared cosine is one. So the entire numerator is minus one, and the answer is minus one over sine squared.

17:10At a half that is minus four point three five zero six eight five. Squeezed against the real quotient: all six bounds cleared, at steps six through eleven. Deepest yet, which is the pattern - the more the shape does, the further in you have to go before it settles. Now flip the numerator's order. The prediction becomes plus four point three five zero six eight five. Same digits. Opposite sign. Zero of the six bounds cleared.

17:43In the product rule the order of the two terms is free. In the quotient rule it is the whole answer. Four rules. Two of them copy the limit rules exactly. Two of them do not, and now you know why. A derivative is a limit of a difference, and the difference of a product splits into two pieces, not one. That is not a mnemonic. It is a single line of algebra, and everything else follows from it.

18:15None of this was proved here, and I want to be straight about that. The rules were stated, and then measured. Eighty-one pairs, thirty-seven places, every rule scored against the definition itself, and the wrong versions scored on the same line so you can see the difference between agreeing and merely answering. Four shapes squeezed rather than differentiated, each one asked how far in it had to go: one step, one step, four, and six.

18:47The one habit worth taking away: when you have an answer, take the function apart a second way and see if you get it again. Ten x said ten both ways. x squared said two x both ways. Two routes, one answer, every time - and if they ever disagree, it is not the mathematics that slipped.