Introduction to Differentiation
Differentiation is a cornerstone of H2 Mathematics and one of the most widely applicable topics in the A-Level syllabus. It measures the rate of change of a function and underpins topics including kinematics, optimisation, curve sketching, and related rates of change.
This guide covers the essential differentiation techniques and their exam applications for H2 Maths students.
Core Differentiation Rules
You must know these rules fluently — they appear in virtually every calculus question:
1. Power Rule
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
Example: d/dx (x⁵) = 5x⁴
2. Chain Rule
Used for composite functions: d/dx [f(g(x))] = f'(g(x)) · g'(x)
Example: d/dx [(3x + 2)⁴] = 4(3x + 2)³ · 3 = 12(3x + 2)³
3. Product Rule
d/dx [uv] = u'v + uv'
Example: Differentiate x² · eˣ → 2x · eˣ + x² · eˣ = eˣ(x² + 2x)
4. Quotient Rule
d/dx [u/v] = (u'v − uv') / v²
Use this when you have a function divided by another function. Many students prefer rewriting as a product (u · v⁻¹) and applying the product rule instead.
Standard Derivatives to Memorise
| Function | Derivative |
|---|---|
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
Applications of Differentiation
1. Gradient of a Curve
f'(x) gives the gradient of the tangent to the curve at any point x. To find the equation of a tangent or normal at a given point:
- Find f'(x) and substitute the x-coordinate to get the gradient m.
- Use point-slope form: y − y₁ = m(x − x₁) for the tangent.
- For the normal, use gradient = −1/m.
2. Stationary Points and Curve Sketching
Stationary points occur where f'(x) = 0. To classify them:
- Find the second derivative f''(x).
- If f''(x) > 0 at the stationary point → minimum.
- If f''(x) < 0 → maximum.
- If f''(x) = 0 → use the first derivative test (check sign change around the point).
3. Optimisation Problems
These are high-value questions in H2 Maths papers. The approach is:
- Define your variables clearly.
- Form an equation for the quantity to be maximised or minimised.
- Use a constraint to reduce to a single-variable function.
- Differentiate, set equal to zero, and solve.
- Verify it is a maximum or minimum using the second derivative.
4. Related Rates of Change
Use the chain rule: dy/dt = dy/dx · dx/dt
Common contexts include expanding circles, filling containers, and shadow length problems. Always draw a diagram and write down all given rates before forming your equation.
Exam Tips for H2 Differentiation
- Always simplify f'(x) before substituting values — it reduces arithmetic errors.
- Show all working for chain, product, and quotient rules — method marks are available.
- For optimisation, always justify your answer (max/min) — don't just state it.
- In related rates questions, clearly define all variables and their units at the start.
Conclusion
Differentiation is a skill that gets sharper with practice. Focus on mastering the core rules first, then apply them systematically to the different application types. Students who are fluent with differentiation find that integration, kinematics, and even statistics topics become significantly more manageable. Invest the time now — it pays off across the entire syllabus.