Three rules for differentiating combinations, the standard derivatives, and what to do when two rates are linked.
Drop a stone into a klong. The ripple's radius grows at a steady half a metre per second. Watch what the area does.
The radius rate never changes. The area rate keeps climbing, because it depends on the radius.
Write the chain before you substitute. Deciding which derivative you want, which one you were given, and what links them is the whole method. Numbers go in last, and a question that asks "at the instant when r = 3" is telling you to substitute at the end, not the start.
| f(x) | sin x | cos x | tan x | ex | ln x | xn |
|---|---|---|---|---|---|---|
| f′(x) | cos x | −sin x | 1cos²x | ex | 1x | nxn−1 |
The scope just widened. At Standard Level the index n had to be a whole number, so √x was out. At Higher Level n can be any fraction, so √x = x½ differentiates to 1⁄2√x, which is ¼ at x = 4. If you learned at SL that roots were not your problem, they are now.
Radians, always. The derivative of sin x is cos x only when x is in radians. A calculator left in degrees will quietly give wrong gradients all the way through a question.
1. Differentiate y = (3x + 1)⁵ and find the gradient at x = 0.
2. Which rule does y = x²e𝕩 need?
3. The ripple's radius grows at 0.5 m per second. Find dAdt when r = 3, to 2 decimal places.
Naming the rule correctly is usually implicit, but applying the quotient rule with the terms the wrong way round is not recoverable: u′v comes first.
On a related rates question, writing the chain of derivatives before any numbers is normally an explicit mark, and it is what makes the rest possible.
Leaving a chain rule half done, differentiating the outside and forgetting to multiply by the inside's derivative, is the single commonest error in this sub-topic. The missing factor is usually a small number like 3, so the answer still looks plausible.
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