Differentiation Rules (Product, Quotient, Chain)

$\frac{dy}{dx}$ is the gradient function, and three rules extend the power rule ($x^n \to nx^{n-1}$) to every function 0606 throws at you. Each rule has a written setup that is both error-prevention and the first method mark.

Product rule, $y = uv$

$(uv)' = u'v + uv'$ $y = x^2 \sin x$: $u = x^2$, $u' = 2x$; $v = \sin x$, $v' = \cos x$ (the setup line) $\frac{dy}{dx} = 2x \sin x + x^2 \cos x$

Quotient rule, $y = \frac{u}{v}$

$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ The numerator’s order is the trap: derivative-of-top times bottom comes FIRST, minus top times derivative-of-bottom. Quote the general formula before substituting, the quoted line survives a sign slip; silent working doesn’t.

$y = \frac{2x + 1}{x^2 + 3}$: $u' = 2$, $v' = 2x$ $\frac{dy}{dx} = \frac{2(x^2 + 3) - (2x + 1)(2x)}{(x^2 + 3)^2} = \frac{6 - 2x - 2x^2}{(x^2 + 3)^2}$

Simplify the numerator; leave the denominator squared and unexpanded, expanding it wastes time and invites errors.

Chain rule, composite functions

For $y = (3x^2 - 5)^4$: outer derivative $\times$ inner derivative:

$\frac{dy}{dx} = 4(3x^2 - 5)^3 \times \mathbf{6x} = 24x(3x^2 - 5)^3$

The forgotten $\times 6x$ is the chain rule’s signature error. Recognition cue: any function of a function. $(\ldots)^n$, $e^{(\ldots)}$, $\ln(\ldots)$, $\sin(\ldots)$, chains. The standard derivatives that feed all three rules (none given in the exam): $\sin x \to \cos x$, $\cos x \to -\sin x$, $\tan x \to \sec^2 x$, $e^x \to e^x$, $\ln x \to \frac{1}{x}$, trig in radians only.

Choosing and combining

Product of two functions $\to$ product rule; ratio $\to$ quotient rule (or rewrite as a product with a negative power); composite $\to$ chain. Hard questions nest them: $y = x^2 e^{3x}$ is a product whose second factor needs the chain. Work outside-in, one rule per line, naming each (“product rule:”, “chain on $e^{3x}$:”), the named lines are followable, and followable is markable.

Common mistakes

• Quotient numerator order flipped
• Chain rule’s inner derivative dropped
• Setup lines skipped, every slip then costs full marks
• Products differentiated term-by-term as if they were sums
• Degree-mode thinking in trig derivatives

Full topic context: Calculus notes · the full exam drill: differentiation technique.