The scope change nobody announces, and why related rates is the half worth the lesson time.
Ask what the area is doing before you press play.
Most will say it grows steadily, because the radius does. Run it and watch dA/dt climb from 1.57 to over 9 while dr/dt never moves off 0.5.
That gap is the chain rule made physical, and it is a far better motivation than a worked example about a bracket raised to a power.
At SL the index n had to be an integer, so root x was out of scope and the SL page says so plainly. Here n is rational, so root x is now routine and examinable.
Say this out loud to anyone who did SL with you. They have been told the opposite, correctly, and nobody will think to tell them it changed.
| Related rates | dA/dr = 2πr, so dA/dt = 2πr × 0.5. At r = 3 that is 3π ≈ 9.42 m² per s. |
| Chain example | (3x + 1)⁵ gives 15(3x + 1)⁴, which is 15 at x = 0 and 3840 at x = 1. |
| Product example | x²e𝕩 gives e𝕩(2x + x²), which is 3e ≈ 8.155 at x = 1. |
| Quotient example | x/(x + 1) gives 1/(x + 1)², which is ¼ at x = 1 and 1 at x = 0. |
| 1. Gradient at x = 0 | 15. |
| 2. Which rule | B, the product rule. |
| 3. dA/dt at r = 3 | 9.42 m² per second. |
1 markWriting the chain of derivatives on a related rates question, before any numbers go in.
1 markApplying the rule correctly, with the quotient rule the right way round: u′v comes first.
1 markSubstituting at the end, at the stated instant, and giving units.
| They give | What it means |
|---|---|
| 5 (Q1) | Differentiated the outside and stopped. The missing inside factor is the commonest error in the whole sub-topic, and because it is usually a small number the answer still looks reasonable. |
| 45 (Q1) | Multiplied by 3 twice. Worth distinguishing from the error above; it is a slip, not a gap. |
| Chain for x²e𝕩 (Q2) | They see two things and reach for the first rule they remember. Ask what is inside what. |
| "None, multiply the derivatives" (Q2) | The instinct that the derivative of a product is the product of the derivatives. Disprove it in ten seconds with x times x. |
| 18.85 (Q3) | Found dA/dr and forgot to multiply by dr/dt. They have the chain but stopped halfway. |
| 28.27 (Q3) | Gave the area, not the rate. Reading rather than mathematics. |
"Which rule do I use?" Make them describe the shape aloud: inside another, side by side, or one over another. The rule follows from the shape and never from the letters.
"Do I need radians?" Yes, and it matters. The derivative of sin x is cos x only in radians. A calculator in degrees gives wrong gradients silently, all the way through.
"Can I use the product rule on a quotient?" Yes, by writing it as a negative power, and sometimes it is quicker. Worth showing once so they know the rules are not a fixed menu.
| What is happening | |
|---|---|
| 1 | The ripple, with no algebra. Why does the area rate change when the radius rate does not? |
| 2 | The chain rule, several single examples, and the missing factor named as the thing to watch. |
| 3 | Product and quotient. Do the quotient slowly; the order and the squared denominator are both losable. |
| 4 | The standard derivatives table, and the n is rational change from SL. |
| 5 | Related rates properly: write the chain, then substitute. |
| 6 | Questions 1 to 3. |
Do not teach related rates as a separate topic at the end of the lesson. It IS the chain rule, and splitting them is why students can do one and not the other.
Do not let them start substituting numbers at the beginning of a related rates question. Values go in last, or the derivative they need has already been turned into a constant.
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.