Differentiation: techniques, stationary points, tangents, normals, rates of change

1 Limit definition of the derivative

2 Notation

• \(y=f(x)\) – function notation.
• \(f'(x)\) – prime notation.
• \(\displaystyle\frac{dy}{dx}\) – Leibniz notation (useful for related‑rates).
• \(dx,\;dy\) – infinitesimal changes (note: \(dx^2\) never appears in the Cambridge syllabus).

3 Fundamental differentiation rules

Rule	Mathematical form	Result
Constant	\(\displaystyle\frac{d}{dx}[c]=0\)	Derivative of a constant is zero.
Power	\(\displaystyle\frac{d}{dx}[x^{n}]=n\,x^{\,n-1}\)	Valid for any real \(n\) (including fractional and negative).
Constant multiple	\(\displaystyle\frac{d}{dx}[c\,f(x)]=c\,f'(x)\)	Factor the constant out.
Sum / difference	\(\displaystyle\frac{d}{dx}[f\pm g]=f'\pm g'\)	Differentiate each term separately.
Product	\(\displaystyle\frac{d}{dx}[f\,g]=f'\,g+f\,g'\)	Differentiate one factor at a time.
Quotient	\(\displaystyle\frac{d}{dx}\!\left[\frac{f}{g}\right]=\frac{f'\,g-f\,g'}{g^{2}}\)	Denominator squared.
Chain	\(\displaystyle\frac{d}{dx}\,f(g(x))=f'\big(g(x)\big)\,g'(x)\)	Derivative of outer function evaluated at inner × derivative of inner.
Implicit differentiation	If \(F(x,y)=0\) then \(\displaystyle\frac{dy}{dx}=-\frac{F_{x}}{F_{y}}\)	Treat \(y\) as a function of \(x\) and differentiate.

4 Derivatives of the core elementary functions (Cambridge list)

Function	Derivative
\(e^{x}\)	\(e^{x}\)
\(a^{x}\;(a>0,\;aeq1)\)	\(a^{x}\ln a\)
\(\ln x\;(x>0)\)	\(\dfrac{1}{x}\)
\(\log_{a}x\;(a>0,\;aeq1)\)	\(\dfrac{1}{x\ln a}\)
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^{2}x\)
\(\csc x\)	\(-\csc x\cot x\)
\(\sec x\)	\(\sec x\tan x\)
\(\cot x\)	\(-\csc^{2}x\)
\(\sin^{-1}x\;( |x|\le1 )\)	\(\dfrac{1}{\sqrt{1-x^{2}}}\)
\(\cos^{-1}x\;( |x|\le1 )\)	\(-\dfrac{1}{\sqrt{1-x^{2}}}\)
\(\tan^{-1}x\)	\(\dfrac{1}{1+x^{2}}\)

5 Stationary points and classification

5.1 Definition

A stationary point occurs where the first derivative vanishes:

\[ f'(x)=0. \]

5.2 Finding stationary points

1. Differentiate \(f(x)\) to obtain \(f'(x)\).
2. Solve \(f'(x)=0\) for the candidate \(x\)-values.
3. Substitute each candidate into \(f(x)\) to get the coordinates \((x,f(x))\).

5.3 Classification – second‑derivative test (Cambridge 1.7.1)

• \(f''(x_0)>0\) → local minimum.
• \(f''(x_0)<0\) → local maximum.
• \(f''(x_0)=0\) → inconclusive – use the first‑derivative sign‑change test or higher‑order derivatives.

5.4 First‑derivative sign‑change test

Examine the sign of \(f'(x)\) just to the left and right of a candidate \(x_0\):

• \(-\) → \(+\) → local minimum.
• \(+\) → \(-\) → local maximum.
• No sign change → point of inflection (may be stationary).

5.5 Example

Find and classify the stationary points of \(f(x)=x^{3}-3x^{2}+2\).

1. \(f'(x)=3x^{2}-6x=3x(x-2)\) → \(x=0,\;2\).
2. Points: \((0,2)\) and \((2,-2)\).
3. \(f''(x)=6x-6\).
    \(f''(0)=-6<0\) → \((0,2)\) is a local maximum.
    \(f''(2)=6>0\) → \((2,-2)\) is a local minimum.

6 Points of inflection

A point of inflection occurs where the concavity changes, i.e. where \(f''(x)\) changes sign.

6.1 Procedure

1. Find \(f''(x)\).
2. Solve \(f''(x)=0\) for possible inflection points.
3. Check the sign of \(f''(x)\) on each side of the candidates.

7 Tangents and normals

7.1 Equation of the tangent

At \((x_0,f(x_0))\) the gradient is \(m=f'(x_0)\). The tangent line is

\[ y-f(x_0)=f'(x_0)\,(x-x_0). \]

7.2 Equation of the normal

The normal is perpendicular to the tangent; its gradient is \(-1/f'(x_0)\) (provided \(f'(x_0)eq0\)).

\[ y-f(x_0)=-\frac{1}{f'(x_0)}\,(x-x_0). \]

7.3 Worked example

For \(y=x^{3}-3x^{2}+2\) at \(x=1\):

• \(f'(1)=3(1)^{2}-6(1)=-3\).
• Point: \((1,0)\).
• Tangent: \(y=-3(x-1)\;\Rightarrow\;y=-3x+3\).
• Normal: \(y=\frac13(x-1)\;\Rightarrow\;y=\frac13x-\frac13\).

8 Related‑rates problems

8.1 General method (Cambridge 1.7.3)

1. Identify all variables and write the geometric/algebraic relation linking them.
2. Differentiate the relation with respect to time \(t\) (use the chain rule where required).
3. Substitute the given numerical values (including known rates) and solve for the required unknown rate.

8.2 Example 1 – Expanding circle

Radius grows at \(\displaystyle\frac{dr}{dt}=0.5\;\text{cm s}^{-1}\). Find \(\displaystyle\frac{dA}{dt}\) when \(r=10\;\text{cm}\).

• Area: \(A=\pi r^{2}\).
• \(\displaystyle\frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\).
• \(\displaystyle\frac{dA}{dt}=2\pi(10)(0.5)=10\pi\;\text{cm}^{2}\text{s}^{-1}\).

8.3 Example 2 – Sliding ladder

A 5 m ladder leans against a wall. The foot slides away at \(\displaystyle\frac{dx}{dt}=1.2\;\text{m s}^{-1}\). Find \(\displaystyle\frac{dy}{dt}\) when the foot is 3 m from the wall.

• Relation: \(x^{2}+y^{2}=5^{2}\).
• Differentiate: \(2x\frac{dx}{dt}+2y\frac{dy}{dt}=0\) → \(\displaystyle\frac{dy}{dt}=-\frac{x}{y}\frac{dx}{dt}\).
• When \(x=3\), \(y=\sqrt{5^{2}-3^{2}}=4\).
   \(\displaystyle\frac{dy}{dt}=-\frac{3}{4}(1.2)=-0.9\;\text{m s}^{-1}\). The top is descending.

9 Basic integration (reverse of differentiation)

9.1 Indefinite integrals – standard forms

Integrand	Integral + C
\(k\) (constant)	\(kx + C\)
\(x^{n}\;(neq-1)\)	\(\dfrac{x^{n+1}}{n+1}+C\)
\(\dfrac{1}{x}\)	\(\ln|x|+C\)
\(e^{x}\)	\(e^{x}+C\)
\(a^{x}\)	\(\dfrac{a^{x}}{\ln a}+C\)
\(\sin x\)	\(-\cos x+ C\)
\(\cos x\)	\(\sin x+ C\)
\(\sec^{2}x\)	\(\tan x+ C\)
\(\csc^{2}x\)	\(-\cot x+ C\)
\(\dfrac{1}{1+x^{2}}\)	\(\tan^{-1}x+ C\)

9.2 Definite integrals – area under a curve

For a continuous function \(f(x)\) on \([a,b]\):

\[ \int_{a}^{b} f(x)\,dx = F(b)-F(a),\qquad F'(x)=f(x). \]

9.3 Volumes of revolution (disc/washer method)

Rotating \(y=f(x)\) about the \(x\)-axis between \(x=a\) and \(x=b\):

\[ V=\pi\int_{a}^{b} \bigl[f(x)\bigr]^{2}\,dx. \]

10 Quadratics (Pure 1)

10.1 Standard forms

• General: \(ax^{2}+bx+c=0\) (\(aeq0\)).
• Completed square: \(a\bigl(x+\tfrac{b}{2a}\bigr)^{2}+k=0\).

10.2 Discriminant

\[ \Delta=b^{2}-4ac\quad\begin{cases} \Delta>0 &\Rightarrow\; \text{two distinct real roots}\\ \Delta=0 &\Rightarrow\; \text{one repeated real root}\\ \Delta<0 &\Rightarrow\; \text{no real roots (complex conjugates)}. \end{cases} \]

10.3 Solving simultaneous linear–quadratic systems

Typical method: substitute the linear expression into the quadratic, solve the resulting quadratic, then back‑substitute.

10.4 Example

Solve \(\begin{cases}y=2x+1\\x^{2}+y^{2}=25\end{cases}\).

• Substitute \(y=2x+1\) into the circle: \(x^{2}+(2x+1)^{2}=25\).
• Expand: \(x^{2}+4x^{2}+4x+1=25\) → \(5x^{2}+4x-24=0\).
• Quadratic formula: \(x=\dfrac{-4\pm\sqrt{4^{2}+480}}{10}=\dfrac{-4\pm22}{10}\).
• Solutions: \(x=1.8,\;y=4.6\) and \(x=-2.6,\;y=-4.2\).

11 Functions, inverses & transformations

• Domain and range – determine by algebraic restrictions (denominators, even roots, logarithms).
• Composite functions: \((f\circ g)(x)=f\bigl(g(x)\bigr)\).
• Inverse function exists iff \(f\) is one‑to‑one; find by swapping \(x\) and \(y\) and solving for \(y\).
• Transformations: \(y=a\,f(bx-c)+d\) – stretch/compress, shift, reflect.

12 Coordinate geometry (Pure 1)

12.1 Straight line

• Slope: \(m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
• Point‑slope form: \(y-y_{1}=m(x-x_{1})\).
• General form: \(Ax+By+C=0\).
• Parallel lines: equal slopes; perpendicular lines: \(m_{1}m_{2}=-1\).

12.2 Distance & midpoint

\[ \text{Distance }AB=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}},\qquad M\Bigl(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\Bigr). \]

12.3 Circle

Standard form: \((x-h)^{2}+(y-k)^{2}=r^{2}\) (centre \((h,k)\), radius \(r\)).

• Tangent at \((x_{0},y_{0})\): \((x_{0}-h)(x-h)+(y_{0}-k)(y-k)=r^{2}\) or \( (x_{0}-h)(x-h)+(y_{0}-k)(y-k)=r^{2}\) simplified to \( (x_{0}-h)(x-h)+(y_{0}-k)(y-k)=r^{2}\).
• Alternative: gradient of radius \(\displaystyle m_{r}=\frac{y_{0}-k}{x_{0}-h}\); gradient of tangent \(m_{t}=-1/m_{r}\).

13 Circular measure (Pure 1)

• Radians: \(2\pi\) rad = \(360^{\circ}\); \(1\text{ rad}= \dfrac{180}{\pi}^{\circ}\).
• Arc length: \(s=r\theta\) (θ in radians).
• Sector area: \(A=\dfrac{1}{2}r^{2}\theta\).

14 Trigonometry (Pure 1 & 2)

14.1 Basic ratios & identities

\[ \sin^{2}\theta+\cos^{2}\theta=1,\qquad 1+\tan^{2}\theta=\sec^{2}\theta,\qquad 1+\cot^{2}\theta=\csc^{2}\theta. \]

14.2 Exact values

For angles \(0^{\circ},30^{\circ},45^{\circ},60^{\circ},90^{\circ}\) (or the radian equivalents) the sine, cosine and tangent values are memorised.

14.3 Sum, difference & double‑angle formulae

\[ \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta,\qquad \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta, \] \[ \sin2\theta=2\sin\theta\cos\theta,\qquad \cos2\theta=\cos^{2}\theta-\sin^{2}\theta. \]

14.4 Solving trig equations

General steps:

1. Rewrite using identities to obtain a single trig function.
2. Isolate the function and use the principal value(s) in the required interval.
3. Include all solutions by adding the appropriate period (\(2\pi\) for sine/cosine, \(\pi\) for tangent).

14.5 Example

Solve \(\;2\sin^{2}\theta-3\sin\theta+1=0\) for \(0\le\theta\le2\pi\).

• Let \(u=\sin\theta\): \(2u^{2}-3u+1=0\) → \((2u-1)(u-1)=0\).
• Solutions: \(u=\tfrac12\) or \(u=1\).
• \(\sin\theta=\tfrac12\) → \(\theta=\frac{\pi}{6},\;\frac{5\pi}{6}\).
   \(\sin\theta=1\) → \(\theta=\frac{\pi}{2}\).

15 Series (Pure 1)

15.1 Arithmetic progression (AP)

\[ a_{n}=a_{1}+(n-1)d,\qquad S_{n}=\frac{n}{2}\bigl(a_{1}+a_{n}\bigr)=\frac{n}{2}\bigl[2a_{1}+(n-1)d\bigr]. \]

15.2 Geometric progression (GP)

\[ a_{n}=a_{1}r^{\,n-1},\qquad S_{n}=a_{1}\frac{1-r^{n}}{1-r}\;(req1),\qquad S_{\infty}= \frac{a_{1}}{1-r}\;( |r|<1). \]

15.3 Example – Sum of a GP

Find the sum of the first 5 terms of \(3,\,6,\,12,\dots\).

• Here \(a_{1}=3,\;r=2\).
• \(S_{5}=3\frac{1-2^{5}}{1-2}=3\frac{1-32}{-1}=93.\)

16 Logarithmic & exponential equations (Pure 2)

• Law of indices: \(a^{m}a^{n}=a^{m+n},\;(a^{m})^{n}=a^{mn},\;a^{m}/a^{n}=a^{m-n}\).
• Log rules (base \(a>0,\;aeq1\)):
   \(\log_{a}(MN)=\log_{a}M+\log_{a}N\),
   \(\log_{a}\dfrac{M}{N}= \log_{a}M-\log_{a}N\),
   \(\log_{a}(M^{k})=k\log_{a}M\),
   \(\log_{a}a=1,\;\log_{a}1=0\).
• Changing base: \(\displaystyle\log_{b}M=\frac{\log_{a}M}{\log_{a}b}\).
• Typical solution method: rewrite the equation so that the same base (or log) appears on both sides, then equate exponents or use the definition of log.

Example

Solve \(2^{x}=5\).

• Take natural logs: \(\ln2^{x}=\ln5\) → \(x\ln2=\ln5\).
• \(x=\dfrac{\ln5}{\ln2}\approx2.322.\)

17 Vectors (Pure 3)

17.1 Notation & basic operations

• Vector \(\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}\) (2‑D: omit the \(k\)-component).
• Magnitude: \(|\mathbf{a}|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}\).
• Scalar (dot) product: \(\mathbf{a}\!\cdot\!\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta\) = \(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\).
• Two vectors are perpendicular iff \(\mathbf{a}\!\cdot\!\mathbf{b}=0\).

17.2 Equation of a line (2‑D)

Through point \(P(x_{1},y_{1})\) with direction vector \(\mathbf{d}=\langle a,b\rangle\):

\[ \frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}. \]

17.3 Example – Angle between vectors

Find the angle between \(\mathbf{u}=\langle2, -1\rangle\) and \(\mathbf{v}=\langle1,3\rangle\).

• \(\mathbf{u}\!\cdot\!\mathbf{v}=2\cdot1+(-1)\cdot3=-1\).
• \(\cos\theta=\dfrac{-1}{\sqrt{5}\sqrt{10}}=-\dfrac{1}{\sqrt{50}}=-\dfrac{\sqrt{2}}{10}\).
• \(\theta\approx95.7^{\circ}\).

18 First‑order separable differential equations (Pure 3)

18.1 General form

\[ \frac{dy}{dx}=g(x)h(y)\quad\Longrightarrow\quad\frac{dy}{h(y)}=g(x)\,dx. \]

18.2 Solution procedure

1. Separate the variables.
2. Integrate both sides (add constant \(C\)).
3. If an initial condition \((x_{0},y_{0})\) is given, substitute to find \(C\).

18.3 Example

Solve \(\displaystyle\frac{dy}{dx}=3x^{2}y^{2}\) with \(y(0)=1\).

• Separate: \(\dfrac{dy}{y^{2}}=3x^{2}\,dx\).
• Integrate: \(-\dfrac{1}{y}=x^{3}+C\).
• Use \(y(0)=1\): \(-1=0+C\) → \(C=-1\).
• Thus \(-\dfrac{1}{y}=x^{3}-1\) → \(y=\dfrac{-1}{x^{3}-1}=\dfrac{1}{1-x^{3}}\).

19 Complex numbers (Pure 3)

19.1 Forms

• Cartesian: \(z=a+bi\) with \(i^{2}=-1\).
• Polar: \(z=r\bigl(\cos\theta+i\sin\theta\bigr)=re^{i\theta}\) where \(r=|z|=\sqrt{a^{2}+b^{2}}\) and \(\theta=\arg z\).

19.2 Operations

• Multiplication: \(r_{1}e^{i\theta_{1}}\cdot r_{2}e^{i\theta_{2}}=r_{1}r_{2}e^{i(\theta_{1}+\theta_{2})}\).
• Division: \(\dfrac{r_{1}e^{i\theta_{1}}}{r_{2}e^{i\theta_{2}}}= \dfrac{r_{1}}{r_{2}}e^{i(\theta_{1}-\theta_{2})}\).
• De‑Moivre’s theorem: \((\cos\theta+i\sin\theta)^{n}= \cos n\theta+i\sin n\theta\).

19.3 Example – Solving \(z^{2}+1=0\)

• Write as \(z^{2}=-1= e^{i\pi}\) (or \(e^{i(\pi+2k\pi)}\)).
• Take square roots: \(z=e^{i(\pi/2+ k\pi)}\) → \(z=\pm i\).

20 Mechanics (Core A‑Level)

20.1 Kinematics (straight‑line motion)

• Velocity: \(v=\dfrac{dx}{dt}\).
• Acceleration: \(a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{dt^{2}}\).
• For constant acceleration: \(v=u+at,\; s=ut+\tfrac12at^{2},\; v^{2}=u^{2}+2as\).

20.2 Forces & Newton’s laws

• Resultant force \(\mathbf{F}=m\mathbf{a}\).
• Weight: \(W=mg\) (downward).
• Friction (kinetic): \(F_{k}=\mu_{k}N\); (static): \(F_{s}\le\mu_{s}N\).

20.3 Work, energy & power

• Work: \(W=\mathbf{F}\!\cdot\!\mathbf{s}=Fs\cos\theta\).
• Kinetic energy: \(E_{k}=\dfrac12mv^{2}\).
• Potential energy (gravity): \(E_{p}=mgh\).
• Power: \(P=\dfrac{dW}{dt}=Fv\) (if force and motion are collinear).

20.4 Momentum

• Linear momentum: \(\mathbf{p}=m\mathbf{v}\).
• Impulse–momentum theorem: \(\displaystyle\int\mathbf{F}\,dt=\Delta\mathbf{p}\).

20.5 Example – Projectile motion (no air resistance)

Launch speed \(u\) at angle \(\theta\) above the horizontal.

• Horizontal: \(x=ut\cos\theta\) (constant velocity).
• Vertical: \(y=ut\sin\theta-\tfrac12gt^{2}\).
• Range: \(R=\dfrac{