Order and degree, and the polynomial condition degree needs before it means anything

0:00Every equation of this kind gets two labels attached to it, and they look like a matched pair. They are nothing of the sort. The first one is always available. You can always find it, on any equation, without asking permission. The second one comes with a condition attached, and when the condition fails, the honest answer is that the label has no value at all. Not zero. Not a half. No value.

0:30So here is one equation with two slots beside it. The first slot always gets filled. The second one sometimes stays empty, and knowing when is most of this topic. The first label is the order, and finding it is a counting job. Look through the equation for every derivative of the unknown. For each one, count how many times the unknown has been differentiated. Keep the largest count. That is the order. It is that mechanical.

1:06A first derivative sitting alone gives order one. A second derivative gives order two. A third gives three. And notice what the count is of. It is of differentiations, not of derivatives. An equation carrying three separate first derivatives is still of order one, because none of them was differentiated more than once. Three instances, worked. A first derivative set equal to an exponential. The deepest derivative present has been taken once, so the order is one.

1:44A second derivative added to the unknown itself. Taken twice. Order two. And a third derivative, added to a coefficient in x times the cube of a second derivative. The deepest is the third one. Order three. Across the twenty-six equations this video works through, every single order came out as written, and every one was found by walking the equation and looking, never by reading an answer off a page.

2:17And that matters more than it sounds. With every expected order raised by one, not a single one of the twenty-six fits. So the orders coming out right is a measurement, and not a count of how many equations are in the list. Now the trap, and it is the expensive one. Look again at that third equation. The third derivative stands there on its own, with no exponent at all. The second derivative is cubed.

2:50The tallest number on the page is that three, sitting on a second derivative. A student scanning the equation for the biggest exponent reads the order as two, because that is where the tall power is. The order is three. A power is not a count of differentiations. It is a count of how many times something is multiplied by itself, which is an entirely different question, and the equation is written the way it is precisely to separate the two.

3:25To make sure that is a real distinction and not a story, a second reader was built that takes the order off whichever derivative carries the tallest power. Across the twenty-six equations, it agrees with the true order seventeen times and disagrees nine times. Seventeen agreements is exactly why the misreading survives. It works most of the time. One more thing the count is not. The count is of derivatives of the unknown. A derivative of some other symbol is not one, however deep it goes.

4:05Take an equation carrying a second derivative of the unknown beside a third derivative of a different letter altogether. The order is two, not three. Swap the two, so that the third derivative is the unknown's, and the order becomes three. Same shapes on the page, different answers, and the only thing that changed was which symbol was being differentiated. The second label is the degree, and before you are allowed to ask for it, the equation has to pass a condition.

4:41The condition is that the equation must be a polynomial in its derivatives. Which means: every derivative of the unknown has to be reachable by adding, by subtracting, by multiplying, and by whole-number powers. Nothing else. And read that clause carefully, because there is a word in it that students drop. In its derivatives. Not in everything. How the unknown itself appears is not being asked about at all. Three test equations make the point.

5:18The first mixes a third derivative, the square of a second, and a first. Everything is added and multiplied and raised to whole powers. It passes. The second squares a first derivative, adds the plain first derivative, and subtracts the square of the sine of the unknown. There is a sine in it. And it passes anyway, because that sine is wrapped around the unknown, not around a derivative. The third adds a first derivative to the sine of a first derivative. That one fails. A derivative is sitting inside a sine, and no amount of adding and multiplying will reach it.

6:02Two admitted, one refused, and the refusal is about where the sine is, not about the sine existing. Once the condition holds, the degree is one specific number. It is the power carried by the derivative that fixed the order. Not the tallest power in the equation. The power on that one derivative. So the arrow runs backwards: first find the order, which tells you which derivative to look at, and then read the power sitting on it.

6:36Go back to the equation with the cube. The order is three, so the derivative to look at is the third one. It carries no exponent, so its power is one. The degree is one. A student reading the tallest exponent answers three and three, and both halves of that answer are wrong, for two different reasons. Every claim in this video was checked twice, and it is worth a minute to say how, because the second check is the interesting one.

7:10The first way is structural. Every equation is written down as a tree, and both labels are computed by walking it. The second way never looks at the tree at all. Pin every symbol in the equation to a number except one derivative. What is left is a function of that derivative alone. Now read that function at equally spaced places and take differences. Then differences of the differences, and so on.

7:42A polynomial of degree d has a difference of order d that is still standing, and a difference of order d plus one that is flat. Nothing else behaves that way. A sine's differences barely shrink. An exponential's differences grow. A square root's shrink far too slowly to ever reach nothing. So the degree can be measured by differencing, without ever asking what functions the equation contains. And an equation the differencing never flattens has no degree.

8:19Nineteen of the twenty-six equations pass the condition. Read at three different sets of pinned numbers, that is fifty-seven readings, and all fifty-seven land on the degree the tree gives. None of the fifty-seven lands on a degree one higher, so the two readers agreeing is not two routines both saying yes to anything. The seven that the tree refuses are refused by the differencing too, at all three pinnings. Twenty-one refusals, and not one number returned.

8:54And the refusal is not the reader running out of room: with the cap raised from eight to sixteen, far past any degree that turns up here, all seven still refuse. Handed a polynomial of degree twelve, the raised cap does answer twelve, so it really is reaching further. The differencing also reports the leading coefficient, not just the degree. On five equations it came out at twenty-four, twenty-four, eight, six and six, each one the degree factorial times the coefficient the equation actually carries.

9:34That last check is what stops an arithmetic slip from hiding behind a degree that happens not to move. One detail in that second reader is not a nicety, and it is worth seeing why. Every derivative in the equation gets asked about, not only the deepest one. Consider the square of a second derivative added to the cosine of a first. Look only at the second derivative, holding the first fixed, and the cosine becomes a constant. The function is a plain square. The differencing happily reports degree two.

10:15But the equation has no degree, because that first derivative is sitting inside a cosine. Asked only about the deepest derivative, the differencing hands a degree to five of the seven equations that have none. A wrapper around a shallow derivative is invisible from the deep derivative's point of view. So you ask about all of them, and one refusal is enough. Here are the verdicts, run down the list.

10:46Of the twenty-six, nineteen have a degree and seven do not. Every one of the seven refusals is named rather than just counted. Four sines, one cosine, one exponential and one root, each with a derivative of the unknown standing inside it. And of the nineteen that do have a degree, not one has a derivative inside any wrapper at all. Now the other direction. Fifteen of the twenty-six carry a sine, a cosine, an exponential, a logarithm or a root somewhere on the page. Eight of those fifteen still have a degree.

11:28So it is not the function that destroys the degree. It is a derivative sitting inside it. And of the nineteen degrees, the tallest power on the page happens to be the right answer ten times, and the wrong answer nine times. Ten out of nineteen is why the habit forms. Both labels, when they exist, are whole numbers greater than zero. Across the whole list, the orders that occur are one, two, three and four. The degrees that occur are one, two and three. Nothing fractional. Nothing zero.

12:07Which rules out the two answers students reach for when an equation resists. The first is a degree of zero, written where not defined belongs. Not defined does not mean zero. It means the question does not arise, and writing zero asserts a whole number that has already been excluded. The second is a fractional degree, from a square root. That second one deserves the last worked example, because it is the one case where the answer changes depending on what you do first.

12:45Take a first derivative set equal to the square root of one plus a second derivative. As it stands, a derivative is sitting under a root, so it is out of a polynomial's reach. No degree. Both readers agree. And if you ignored the condition and simply multiplied the powers out, you would report a degree of one half, which is a number the rules have already excluded. Now square both sides. The first derivative squared equals one plus the second derivative.

13:20The root is gone. The equation is a polynomial in its derivatives. Order two, degree one, and again both readers agree. And the two really are the same equation. At six different values of the second derivative, squaring the root returns one plus that second derivative, to within a unit in the twentieth decimal place. So clearing a root is fair, and it changed the degree verdict without touching the order, which was two before and two after.

13:54What is not fair is dividing by a derivative, or taking a root of both sides to make something look tidier. Clear roots and denominators, and then stop. So, the whole of it. Order is a count of differentiations, taken on derivatives of the unknown, and a tall power on a shallow derivative does not raise it. Degree is conditional. The equation has to be a polynomial in its derivatives first, and how the unknown itself appears is not part of that question.

14:33When degree is allowed, it is the power on the derivative that fixed the order, and not the tallest power on the page. When it is not allowed, the answer is that it is not defined, with the reason given: a derivative is standing inside something a polynomial cannot reach. And both labels, whenever they exist, are whole numbers above zero. One of them is a count you can always take. The other is a measurement that first has to be earned.