Differentiation: further techniques, higher derivatives, stationary points

1️⃣ Further Differentiation Techniques

Think of differentiation like a toolbox. Each rule is a special tool that helps you break down complex expressions into simpler parts.

• Product Rule – For multiplying two functions:
  $f(x)=u(x)v(x)\;\;\Rightarrow\;\;f'(x)=u'(x)v(x)+u(x)v'(x)$
• Quotient Rule – For dividing two functions:
  $f(x)=\dfrac{u(x)}{v(x)}\;\;\Rightarrow\;\;f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2}$
• Chain Rule – For nested functions:
  $f(x)=g(h(x))\;\;\Rightarrow\;\;f'(x)=g'(h(x))\,h'(x)$
• Logarithmic Differentiation – Useful when a function is a product or power of many terms. Take $\ln$ of both sides, differentiate, then exponentiate back.
• Trigonometric Differentiation – Remember the patterns: $\dfrac{d}{dx}\sin x=\cos x$, $\dfrac{d}{dx}\cos x=-\sin x$, $\dfrac{d}{dx}\tan x=\sec^2 x$, etc.
• Inverse Function Rule – If $y=f^{-1}(x)$, then $y'=\dfrac{1}{f'(y)}$.

Example: Differentiate $f(x)=x^2\sin x$ using the product rule.

$f'(x)=2x\sin x + x^2\cos x$

2️⃣ Higher Derivatives

Just like you can take the first derivative (rate of change), you can keep differentiating to find higher rates of change.

Notation: $f^{(n)}(x)$ is the $n^{th}$ derivative. For example, $f^{(2)}(x)=f''(x)$, $f^{(3)}(x)=f'''(x)$.

Why care? Higher derivatives appear in Taylor series, curvature, and acceleration in physics.

Example: Find the third derivative of $f(x)=x^4$.

$f'(x)=4x^3$
$f''(x)=12x^2$
$f'''(x)=24x$

Notice the pattern: each differentiation reduces the exponent by 1 and multiplies by the previous exponent.

3️⃣ Stationary Points & Inflection Points 🔍

A stationary point is where the slope is zero: $f'(x)=0$. Think of a hilltop or valley where the road momentarily stops going up or down.

First Derivative Test – Check the sign of $f'$ just before and after the point.

• If $f'$ changes from + to -, the point is a local maximum.
• If $f'$ changes from - to +, the point is a local minimum.
• If the sign doesn't change, it's a point of inflection (the curve just flattens out).

Second Derivative Test – If $f''(x)eq0$ at a stationary point:

• $f''(x)>0$ → local minimum.
• $f''(x)<0$ → local maximum.

Example: Find stationary points of $f(x)=x^3-3x^2+2x$.

$f'(x)=3x^2-6x+2=0$ → $x=\dfrac{6\pm\sqrt{36-24}}{6}=\dfrac{6\pm2}{6}=\{1,\; \tfrac{2}{3}\}$
$f''(x)=6x-6$ → $f''(1)=0$ (inflection), $f''(\tfrac{2}{3})=-2$ (maximum).
So, $x=\tfrac{2}{3}$ is a local maximum, $x=1$ is a point of inflection.

Remember: stationary points are where the graph “turns” or “flattens” – they’re key for sketching graphs and solving optimisation problems.

📝 Examination Tips

• Always show your work – examiners look for clear steps, not just the final answer.
• Use the first derivative test when the second derivative is messy or zero.
• When asked for a higher derivative, write $f^{(n)}(x)$ and simplify step by step.
• Check units: if you’re dealing with physical quantities, keep track of dimensions.
• Practice converting between product/quotient forms and logarithmic differentiation – it saves time.