Pure Mathematics 3 (P3) Calculus Cheat
Sheet
1. Advanced Differentiation
Core Formulas & Mechanics
● Implicit Differentiation: Differentiate term-by-term with respect to $x$. When
differentiating a term containing $y$, apply the chain rule:
$$\frac{d}{dx}\big(f(y)\big) = f'(y) \cdot \frac{dy}{dx}$$
Example: $\frac{d}{dx}(x^2 y^3) = 2x y^3 + 3x^2 y^2 \frac{dy}{dx}$ (Product rule +
Implicit).
● Parametric Differentiation:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
$$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$$
Mark-Scheme Traps
● Stationary Points on Parametric Curves: Set $\frac{dy}{dt} = 0$ for horizontal
tangents ($\frac{dx}{dt} \neq 0$). Set $\frac{dx}{dt} = 0$ for vertical tangents.
● Exact Values: Never write decimals unless explicitly asked. Leave answers in exact
form (e.g., $\frac{2}{3}\ln 5$ or $\sqrt{3}e^2$).
2. Advanced Integration
Core Patterns & Techniques
A. Reverse Chain Rule
● $\int \frac{f'(x)}{f(x)} \, dx = \ln\vert{}f(x)\vert{} + C$
● $\int f'(x)[f(x)]^n \, dx = \frac{[f(x)]^{n+1}}{n+1} + C \quad (n \neq -1)$
B. Integration by Parts
$$\int u \frac{dv}{dx} \, dx = uv - \int v \frac{du}{dx} \, dx$$
● LATE Priority for choosing $u$: Logarithms, Algebraic, Trigonometric,
Exponentials.
● Note: To integrate $\ln x$, treat it as $\int 1 \cdot \ln x \, dx$ where $u = \ln x$ and
$\frac{dv}{dx} = 1$.
C. Partial Fractions Integration
Convert $\frac{P(x)}{Q(x)}$ into partial fractions before integrating:
$$\int \frac{5x - 1}{(x+1)(x-2)} \, dx = \int \left( \frac{2}{x+1} + \frac{3}{x-2} \right) dx =
2\ln\vert{}x+1\vert{} + 3\ln\vert{}x-2\vert{} + C$$
D. Trigonometric Integration Essentials
● $\int \tan x \, dx = \ln\vert{}\sec x\vert{} + C$
● $\int \sec x \, dx = \ln\vert{}\sec x + \tan x\vert{} + C$
● Use $\cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$ to integrate $\cos^2 x$ or $\sin^2 x$.
3. Differential Equations & Word Problems
Execution Steps
1. Separate Variables: Group all $y$ and $dy$ terms on one side, and all $x$ and
$dx$ terms on the other:
$$\frac{dy}{dx} = f(x)g(y) \implies \int \frac{1}{g(y)} \, dy = \int f(x) \, dx$$
2. Integrate both sides independently: Add $+C$ on the right side.
3. Find $C$ early: Substitute given initial boundary conditions $(x_0, y_0)$ as soon as
integration is finished before rearranging for $y$.
Translating Word Problems into Differential Equations
● "Rate of change of $P$ is proportional to $P$":
$$\frac{dP}{dt} = kP$$
● "Rate of decrease of temperature $T$ is proportional to the difference between $T$
and room temperature $T_0$":
$$\frac{dT}{dt} = -k(T - T_0)$$
4. Binomial Series & Vector Essentials (P3 Flash-Notes)
Binomial Expansion for Rational Indices $(1 + x)^n$ where $n \notin
\mathbb{N}$
$$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \dots$$
● Validity Condition: Valid only if $\vert{}x\vert{} < 1$.
● If expanding $(a + bx)^n$, factor out $a^n$ first:
$$(a + bx)^n = a^n \left(1 + \frac{b}{a}x\right)^n \quad \text{valid for }
\left\vert{}\frac{b}{a}x\right\vert{} < 1$$
Vector Line Equations
$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$
● $\mathbf{a}$ = position vector of a point on the line.
● $\mathbf{b}$ = direction vector of the line.
● Angle between two lines: Use scalar product of their direction vectors
($\mathbf{b}_1 \cdot \mathbf{b}_2 =
\vert{}\mathbf{b}_1\vert{}\vert{}\mathbf{b}_2\vert{} \cos\theta$).
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