H2 Maths Notes (JC 1-2): 5.1) Differentiation

> **Before you revise**\\
> Memorise derivative rules with the full statement (chain, product, quotient). Practise switching between coordinate, parametric, and implicit forms so you can differentiate anything the paper throws at you.

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## Core Derivative Rules

- Power rule: \\( \frac\{d}\{dx} x^n = nx^\{n-1} \\) for rational \\( n \\).
- Product rule: \\( \frac\{d}\{dx}\[uv] = u'v + uv' \\).
- Quotient rule: \\( \frac\{d}\{dx}\left( \frac\{u}\{v} \right) = \frac\{u'v - uv'}\{v^2} \\).
- Chain rule: \\( \frac\{d}\{dx} f(g(x)) = f'(g(x)) g'(x) \\).
- Exponential and logarithmic derivatives: \\( \frac\{d}\{dx} e^{kx} = k e^{kx} \\), \\( \frac\{d}\{dx} \ln x = \frac\{1}\{x} \\).

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## Trigonometric and Hyperbolic Functions

- \\( \frac\{d}\{dx} \sin x = \cos x \\), \\( \frac\{d}\{dx} \cos x = -\sin x \\), \\( \frac\{d}\{dx} \tan x = \sec^2 x \\).
- Inverses: \\( \frac\{d}\{dx} \sin^\{-1} x = \frac\{1}\{\sqrt\{1 - x^2}} \\), etc.
- Hyperbolic: \\( \frac\{d}\{dx} \sinh x = \cosh x \\), \\( \frac\{d}\{dx} \cosh x = \sinh x \\).

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## Implicit Differentiation

- Differentiate both sides treating \\( y \\) as function of \\( x \\) and solve for \\( \frac\{dy}\{dx} \\).

**Example -- Implicit derivative**

Given \\( x^2 + xy + y^2 = 7 \\).

1. Differentiate: \\( 2x + y + x \frac\{dy}\{dx} + 2y \frac\{dy}\{dx} = 0 \\).
2. Factor: \\( \left( x + 2y \right) \frac\{dy}\{dx} = -(2x + y) \\).
3. Hence \\( \frac\{dy}\{dx} = -\frac\{2x + y}\{x + 2y} \\).

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## Parametric Differentiation

- For \\( x = f(t) \\), \\( y = g(t) \\): \\( \frac\{dy}\{dx} = \frac\{\frac\{dy}\{dt}}\{\frac\{dx}\{dt}} \\).
- Second derivative: \\( \frac\{d^2 y}\{dx^2} = \frac\{1}\{\frac\{dx}\{dt}} \frac\{d}\{dt}\left( \frac\{dy}\{dx} \right) \\).

**Example -- Parametric slope**

Given \\( x = t^2 + 1 \\), \\( y = \ln(1 + t) \\).

1. \\( \frac\{dx}\{dt} = 2t \\, \frac\{dy}\{dt} = \frac\{1}\{1 + t} \\).
2. \\( \frac\{dy}\{dx} = \frac\{1}\{2t(1 + t)} \\).

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## Tangents and Normals

- Tangent gradient \\( m_t = \frac\{dy}\{dx} \\) at point; normal gradient \\( m_n = -\frac\{1}\{m_t} \\).
- Tangent line: \\( y - y_0 = m_t (x - x_0) \\).
- Normal line: \\( y - y_0 = m_n (x - x_0) \\).

**Example -- Tangent to curve**

Find tangent at \\( x = 1 \\) for \\( y = x e^{-x} \\).

1. \\( \frac\{dy}\{dx} = e^{-x} - x e^{-x} = e^{-x}(1 - x) \\).
2. At \\( x = 1 \\), gradient zero → tangent horizontal: \\( y = \frac\{1}\{e} \\).

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## Optimisation and Stationary Points

- Stationary points satisfy \\( \frac\{dy}\{dx} = 0 \\); classify with second derivative test.
- Second derivative: \\( \frac\{d^2 y}\{dx^2} > 0 \\) (minimum), \\( \< 0 \\) (maximum), zero → investigate further.
- For optimisation, interpret result in context (length, area, cost).

**Example -- Optimisation**

Minimise surface area of open-top box with base square of side \\( x \\) and volume \\( 108 \pu\{cm3} \\).

1. Height \\( h = \frac\{108}\{x^2} \\).
2. Surface area \\( S = x^2 + 4xh = x^2 + \frac\{432}\{x} \\).
3. Differentiate: \\( \frac\{dS}\{dx} = 2x - \frac\{432}\{x^2} = 0 \Rightarrow 2x^3 = 432 \\).
4. \\( x = \sqrt\[3]\{216} = 6 \\), \\( h = 3 \\).

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## Related Rates

- Differentiating implicit relationships with respect to time: \\( \frac\{dz}\{dt} = \frac\{dz}\{dx} \frac\{dx}\{dt} \\).

**Example -- Expanding circle**

A circle radius \\( r \\) grows at \\( \frac\{dr}\{dt} = 0.2 \pu\{cm.s-1} \\). Find rate of change of area when \\( r = 5 \pu\{cm} \\).

1. Area \\( A = \pi r^2 \\).
2. \\( \frac\{dA}\{dt} = 2\pi r \frac\{dr}\{dt} = 2\pi \times 5 \times 0.2 = 2\pi \pu\{cm2.s-1} \\).

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## Calculator Workflow

- GC differentiation function evaluates derivatives numerically; record inputs (e.g. `d/dx` at specific points).
- Use `TABLE` to evaluate derivative sign around stationary points for classification.
- Store intermediate expressions to avoid algebra slips when differentiating complex functions.

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## Exam Watch Points

- Present exact derivatives before substituting numerical values.
- State the method used (implicit, parametric) and justify each step clearly.
- Include units in related-rates answers.
- For optimisation, verify solutions satisfy constraints (positive dimensions, etc.).

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## Quick Revision Checklist

- \[ ] Apply product, quotient, and chain rules without hesitation.
- \[ ] Differentiate implicit and parametric relations, reporting ( \frac\{dy}\{dx} ) clearly.
- \[ ] Evaluate tangents, normals, and stationary points with proper classification.
- \[ ] Tackle optimisation and related-rate problems with a structured plan and unit-aware answers.

Next steps: Move to Topic 5.2 for Maclaurin series expansion techniques.