The scope limit that matters more than the rule, and the subtraction slip that produces most of the wrong answers.
On Applications and Interpretation the index is an integer. Negative powers are in scope. Fractional powers are not, so √x is not something an AI student is asked to differentiate.
Analysis and Approaches has a rational index and does differentiate √x as routine. Nearly every textbook, worksheet and video you will find treats root x as a standard case, so an AI class taught from AA material spends time on something that cannot be set for them, and often meets it before they are secure on the integer cases that can.
Question 3 on the student page tests exactly this, deliberately. It is the one question on the page that is about the course rather than the mathematics.
The three-move reveal. Multiply by the power, then knock one off it, in that order. Play it, then make the class say the two moves back to you before anything else happens. Order matters: students who drop the power first then multiply by the new one get the wrong coefficient.
The rule against a real chord. This is the one to use if anyone asks why the rule works, and it is worth using even if nobody asks. One readout is the rule, the other is a genuine chord across a gap of 0.0001, computed the 5.1 way. They agree to three decimals at every x. Without this, the power rule arrives one lesson after you defined a gradient as a limit, and the limit quietly becomes something they sat through.
| The worked quartic | f(x) = 3x⁴ − 5x² + 7x − 2 gives f′(x) = 12x³ − 10x + 7. |
| 1. f′(1) | 9. 12 − 10 + 7. For reference f(1) = 3, f′(0) = 7 and f′(2) = 83. |
| 2. Differentiate 4/x² | A. −8x−3, which is −8/x³. At x = 2 that is −1. |
| 3. Not expected to differentiate | B, √x, because the index is not a whole number. |
| Check widget | g(x) = 2x³ + x² − 4x + 9, g′(x) = 6x² + 2x − 4. g′(1) = 4 and g′(−1) = 0. |
1 markRewriting a fraction or a root as a power before differentiating. Often an explicit method mark, and it is the step that makes the rest possible.
1 markThe derivative itself, term by term, with signs carried through.
1 markSubstituting correctly afterwards, when the question asks for a value. Differentiating and then putting x into the original function is a surprisingly frequent way to lose it.
| They write | What it means |
|---|---|
| 3 for f′(1) | They substituted into f, not f′. Name the two functions separately on the board every time. |
| 7 for f′(1) | Only the constant term of the derivative. They stopped early, usually because 7x looked like the last "real" term. |
| −8x−1 | The big one. Knocking one off minus 2 and getting minus 1. A subtraction error, not a calculus error, and it needs naming as such or they look for the fault in the wrong place. |
| 8x−3 | Lost the sign when multiplying by a negative power. Right structure. |
| −2x−3 | Used the power as the new coefficient and dropped the 4. They have half-remembered the rule as "the power comes to the front" rather than "multiply by the power". |
| Chose x−3 on Q3 | They think negative means out of scope. Reassure: negative whole numbers are fine, fractions are not. |
"Where did the minus 2 go?" It differentiated to zero. Say that it is something they did, not something they skipped, and show two parabolas differing only by a constant: same shape, same gradient everywhere.
"Why is the derivative of 7x just 7?" Because 7x is a straight line of gradient 7, and a line has the same gradient everywhere. The rule agrees: 7x¹ gives 1 × 7 × x⁰, and x⁰ = 1.
"Can I just use my calculator?" For a numerical derivative at a point, often yes, and they should know how. For "find f′(x)" they need the algebra, and the method marks are in the working.
| What is happening | |
|---|---|
| 1 | Recall 5.1: a gradient was a limit from a table. Ask whether they want to do that every time. They do not. Motivate the rule. |
| 2 | The three-move reveal, then the two moves said back to you. Several single terms on the board together. |
| 3 | The term-by-term animation, then a polynomial of their own. |
| 4 | The check widget. Why the rule is not a magic trick. |
| 5 | Negative powers. Rewrite first, every time. The minus 2 to minus 3 step on the board, slowly. |
| 6 | Questions 1 to 3. Question 3 is a conversation about their course, not a calculation. |
| 7 | Set practice that includes at least one fraction to rewrite. |
Do not say "bring the power down and reduce it by one" without making the order explicit. Students who reduce first and then multiply by the new power get 9x³ from 3x⁴ and cannot see why.
Do not demonstrate √x "for interest" in an AI class. It reads as examinable, it is the first thing they will practise, and the integer cases are not secure yet.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
The reveals respect a reduced-motion preference: if the device asks for less animation, every row is shown at once and the button reads "Shown". Nothing is loaded from anywhere else and nothing is stored.