Choosing the rule from the shape, and the index that quietly became a fraction.
Make them check one derivative before they trust any of them.
The checker compares the rule against a gradient measured from a tiny chord, live, at any x. Use it once on a function the class has just differentiated and the habit lands.
It also quietly reconnects this sub-topic to 5.1, where a gradient was defined as a limit. Without that link the three rules arrive as unexplained machinery.
At 5.3 the index n had to be an INTEGER. Here it is rational, so √x is in scope and differentiates to 1/(2√x), which is ⅝ at x = 9.
Say it out loud. Students who were correctly told at 5.3 that roots were not their problem have no reason to think it changed, and the first root they meet is usually in an exam.
| Chain | sin(3x − 1) gives 3cos(3x − 1), which is 1.6209 at x = 0. ex²+2 gives 2xex²+2, which is 2e³ ≈ 40.171 at x = 1. |
| Product | x²sin x gives 2x sin x + x²cos x, which is π at x = π/2. |
| Quotient | (sin x)/x gives (x cos x − sin x)/x², which is −1/π at x = π. |
| 1. Root x at x = 9 | 0.1667, from 1/(2√x). |
| 2. Which rule | C, the quotient rule, though rewriting as a product also works. |
| 3. sin(3x − 1) at x = 0 | 1.6209. |
1 markRewriting roots and fractions as powers before differentiating.
1 markThe inside derivative on a chain rule, which is the mark most often lost.
1 markQuotient rule in the right order: u′v first.
| They give | What it means |
|---|---|
| 0.5403 (Q3) | Chain rule left half done: the cos is right, the factor of 3 is missing. The commonest error in the sub-topic. |
| 2.9996 (Q3) | Calculator in degrees. Worth checking across the room the first time trigonometry is differentiated. |
| 3 for root x at 9 | Gave √9, the y value, not the gradient. |
| Product rule for a fraction | They have not registered that a rewrite is available. Not wrong, but slower. |
| Derivatives multiplied | The belief that (uv)′ = u′v′. Disprove with x times x in ten seconds. |
"How do I know which rule?" Make them say the shape aloud: inside, beside, or over. The rule follows from the shape, never from the letters.
"Do I need radians?" Yes, and a calculator in degrees produces wrong gradients silently for a whole question.
"Can I use the product rule instead of the quotient rule?" Yes, by writing the bottom as a negative power, and it is often quicker. Show it once so the rules feel like tools rather than a menu.
| What is happening | |
|---|---|
| 1 | Why the power rule alone is not enough. Three functions it cannot touch. |
| 2 | The chain rule, with the missing factor named as the thing to watch. |
| 3 | Product and quotient, the quotient slowly. |
| 4 | The standard derivatives, and the rational index change from 5.3. |
| 5 | The checker, then questions 1 to 3. |
| 6 | Mixed practice where they must choose the rule, not be told it. |
Do not give a worksheet where every question uses the same rule. Choosing is the skill, and a page of chain rule practice does not build it.
Do not skip the radians point because it feels obvious. It is the single most expensive silent error in the topic.
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.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.