Chain, product and quotient rules (AA SL)

Everything so far has been one power at a time. Real functions are built by nesting, multiplying and dividing, and each of those has its own rule. The hard part is not the rules. It is deciding which one you are looking at.

inside→ chain One function wrapped in another, like sin(3x − 1). Differentiate the outside, keep the inside, multiply by the inside's derivative.

beside→ product Two functions multiplied, like x²sin x. Each takes a turn: u′v + uv′.

over→ quotient One divided by another. Top derivative first, all over the bottom squared.

inside → chain · beside → product · over → quotient

Read the shape first. The letters tell you nothing.

Check any derivative you produce

Pick a function and move x. The left readout is the rule applied; the right is the gradient measured from a tiny chord, the way 5.1 defined it. If your rule is right they agree everywhere.

y = sin(3x − 1)

x = 0.50

–from the rule

–from a tiny chord

they agreeverdict

The standard derivatives

f(x)	xⁿ	sin x	cos x	e^x	ln x
f′(x)	nxⁿ⁻¹	cos x	−sin x	e^x	1x

n is now a fraction, not just a whole number. At 5.3 the index had to be an integer. Here it does not, so √x = x^½ differentiates to ¹⁄_2√x, which is ⅝ at x = 9. Rewrite every root and every fraction as a power before you start.

Radians only. The derivative of sin x is cos x only when x is in radians. A calculator left in degrees produces wrong gradients silently, through an entire question.

One of each

sin(3x − 1)→ cos(3x − 1) × 3 = 3cos(3x − 1) chain. At x = 0 that is 3cos(−1) ≈ 1.6209 e^x²+2→ e^x²+2 × 2x = 2xe^x²+2 chain. At x = 1 that is 2e³ ≈ 40.171 x²sin x→ 2x sin x + x²cos x product. At x = π/2 that is π sin xx→ x cos x − sin xx² quotient. At x = π that is −1/π

Your turn

1. Differentiate y = √x and find the gradient at x = 9. Give it as a decimal to 4 places.

2. Which rule does y = e^xx² need first?

The chain rule. The product rule. The quotient rule. None of them.

3. Differentiate y = sin(3x − 1). What is the gradient at x = 0, to 4 decimal places?

Where the marks go

Rewriting roots and fractions as powers before differentiating. It is often an explicit method mark and it is what makes the rest possible.

The inside derivative on a chain rule. Forgetting it is the commonest error on this sub-topic, and because the missing factor is usually a small number the answer still looks plausible.

Quotient rule order: u′v comes first. Getting it backwards flips the sign of the whole answer.

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