Differentiation: Product, Quotient and Chain Rule

Paper 1 routinely bundles the three differentiation rules into one multi-part question. Each part is short, but each has a characteristic finishing step, factorising a common bracket (product), combining into one fraction (quotient), and not forgetting the inner derivative (chain).

Differentiate each of the following with respect to $x$. (i) $y = x^2\sqrt{2x + 1}$ [3] (ii) $y = \dfrac{x}{x^2 + 1}$ [3] (iii) $y = (4x - 3)^6$ [2]

The working

(i) Product rule with $u = x^2$ and $v = (2x+1)^{1/2}$. $u' = 2x, \qquad v' = \tfrac{1}{2}(2x+1)^{-1/2}\cdot 2 = (2x+1)^{-1/2} \quad \text{(M1 for chain on } v)$ $\frac{dy}{dx} = u'v + uv' = 2x(2x+1)^{1/2} + x^2(2x+1)^{-1/2} \quad \text{(M1)}$

Factor out $(2x+1)^{-1/2}$ to reach the expected single-fraction form: $= (2x+1)^{-1/2}\big[2x(2x+1) + x^2\big] = \frac{4x^2 + 2x + x^2}{\sqrt{2x+1}} = \frac{5x^2 + 2x}{\sqrt{2x+1}} \quad \text{(A1)}$

(ii) Quotient rule with $u = x$, $v = x^2 + 1$: $\frac{dy}{dx} = \frac{u'v - uv'}{v^2} = \frac{(1)(x^2+1) - x(2x)}{(x^2+1)^2} \quad \text{(M1, M1)}$ $= \frac{x^2 + 1 - 2x^2}{(x^2+1)^2} = \frac{1 - x^2}{(x^2+1)^2} \quad \text{(A1)}$

(iii) Chain rule, power outside then derivative of the inside: $\frac{dy}{dx} = 6(4x - 3)^5 \cdot 4 = 24(4x - 3)^5 \quad \text{(M1, A1)}$

Where the marks are won and lost

• In (i) the factorising step is the accuracy mark. Leaving $2x\sqrt{2x+1} + \frac{x^2}{\sqrt{2x+1}}$ is correct but often only scores the method marks unless the question says “you may leave your answer unsimplified”.
• In (ii) the quotient rule sign is $u'v \textbf{ minus } uv'$. Reversing it flips the sign of the whole numerator.
• In (iii) the "$\times 4$" is the chain-rule inner derivative and a mark in its own right. $6(4x-3)^5$ alone is incomplete.

Common mistakes

• Writing the quotient rule as $\frac{uv' - u'v}{v^2}$ (order reversed).
• Dropping the inner derivative in the chain rule (the $\times 2$ in (i), the $\times 4$ in (iii)).
• Mishandling the negative fractional power $(2x+1)^{-1/2}$ when factorising.

Full method: Differentiation Rules notes. Topic home: Calculus pillar.