Calculus

Calculus is the summit of the 0606 syllabus, the largest topic, the most marks, and the content that carries furthest into A Level. It splits into differentiation (with four applied question families) and integration (with two). Each family has a fixed exam routine; the two dedicated technique guides, differentiation and integration & area, drill them. These notes are the map.

Differentiation: the rules

$\frac{dy}{dx}$ is the gradient function. The core kit (none of it given in the exam, memorise list):

• Power rule: $x^n \to nx^{n-1}$
• Product rule: $(uv)' = u'v + uv'$
• Quotient rule: $(u/v)' = \frac{u'v - uv'}{v^2}$
• Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, for composites like $(3x^2 - 5)^4$, $e^{2x}$, $\ln(x^2 + 1)$
• Standard derivatives: $\sin x \to \cos x$, $\cos x \to -\sin x$, $\tan x \to \sec^2 x$, $e^x \to e^x$, $\ln x \to \frac{1}{x}$ (radians assumed for trig)

The universal working habit: write the setup ($u$, $u'$, $v$, $v'$ or the chain decomposition) before differentiating. That line carries the method mark and prevents the sign slips that dominate examiner reports.

The four differentiation question families

1. Tangents and normals. Differentiate $\to$ substitute $x$ $\to$ tangent gradient $m$ (normal: $-\frac{1}{m}$) $\to$ line equation $y - y_1 = m(x - x_1)$. Read the question twice: tangent or normal?
2. Stationary points. Set $\frac{dy}{dx} = 0$ (write the line, it’s a mark), solve, classify with $\frac{d^2y}{dx^2}$: negative $\to$ maximum, positive $\to$ minimum, stating the conclusion in words. Applied max/min (“find the dimensions minimising the area”) add a modelling step: build the function, then run the routine.
3. Rates of change. Connected rates via the chain: $\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$. State the chain before substituting numbers, the stated chain is the method mark.
4. Small changes/approximations where included: $\delta y \approx \frac{dy}{dx} \cdot \delta x$.

Integration: the reverse

$\int x^n \, dx = \frac{x^{n+1}}{n+1}$ $+ c$ ($n \ne -1$), plus the bracket forms $\int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c$ and the standard integrals of $e^{ax+b}$, $\sin(ax+b)$, $\cos(ax+b)$, each “divide by the inside coefficient”. The dropped $+c$ is the syllabus’s most predictable lost mark. Finding $c$ from a given point (“the curve passes through $(1, 3)$”) is its own question family: integrate, substitute, solve.

Definite integrals: write the bracketed antiderivative, then the substituted line $F(b) - F(a)$ before evaluating, visible substitution is where the method credit sits, especially on Paper 1 where answers land as exact fractions.

Area under a curve

The four-step routine: sketch $\to$ identify the region $\to$ set up the integral(s) $\to$ evaluate with visible substitution. Below-axis regions integrate negative (split and take magnitudes); curve-and-line regions use $\int (\text{top} - \text{bottom})$ between intersection points. The full set of traps and variants is in the integration technique guide.

Kinematics: calculus in motion

For a particle on a line with displacement $s(t)$:

$v = \frac{ds}{dt}$, $a = \frac{dv}{dt}$, differentiate forward; integrate to go back, constants from initial conditions

The translation dictionary does the marking work: at rest $\to v = 0$; at the origin/O $\to s = 0$; velocity is minimum $\to \frac{dv}{dt} = 0$; total distance (vs displacement) $\to$ check for direction changes ($v = 0$) inside the interval and integrate $|v|$ piecewise. Each translated statement, written down, is typically a mark.

Worked exam-style question

A curve has equation $y = x^3 - 6x^2 + 9x + 2$. Find the stationary points and determine their nature.

$\frac{dy}{dx} = 3x^2 - 12x + 9$ (M, A) $\frac{dy}{dx} = 0$: $3(x^2 - 4x + 3) = 0 \to 3(x - 1)(x - 3) = 0 \to x = 1, 3$ (M, A) Points: $(1, 6)$ and $(3, 2)$ (A) $\frac{d^2y}{dx^2} = 6x - 12$. At $x = 1$: $-6 < 0 \to$ $(1, 6)$ is a maximum. At $x = 3$: $+6 > 0 \to$ $(3, 2)$ is a minimum (M, A, with conclusions in words)

Seven marks; every one attached to a written line. This question, in some costume, appears in essentially every session.

Common mistakes in this topic

• Quotient-rule numerator order flipped; chain rule’s inner derivative forgotten
• "$\frac{dy}{dx} = 0$" used but never written, the silent mark donation
• $+c$ dropped; initial conditions never applied in kinematics
• Tangent given where the normal was asked
• Degrees used in trig calculus (radians always, see circular measure)
• Distance/displacement conflated when velocity changes sign

Calculus gets weeks 1–2 of the revision plan because mastery here moves more marks than anywhere else, and it’s where an A* is genuinely built.

Calculus is also where 1-to-1 marking feedback compounds fastest. Teacher Rig has marked eight years of these scripts, free 1-hour trial class, booked on WhatsApp.