Grade 12 · CAPS-aligned
Grade 12 Calculus — Differentiation from First Principles to Cubic Graphs
Learn the rules, not the magic: first principles, derivatives, tangents and cubic graphs.
Calculus is a major Grade 12 Paper 1 area. The good news: many marks come from routine differentiation, tangents, cubic-graph interpretation and optimisation when your method is clear.
Why most learners find this hard
First principles intimidates learners because of the limit notation, but it's actually formulaic. The optimisation word problems are where real marks are lost.
What you'll learn
- First-principles definition of f′(x)
- The power rule: d/dx (xⁿ) = n·xⁿ⁻¹
- Finding gradients at a point and equations of tangents
- Sketching cubic graphs: intercepts, stationary points, inflection
- Optimisation: maximising area / minimising cost
Worked example
Find f′(x) if f(x) = x³ − 6x² + 9x.
- 1Differentiate term-by-term using the power rule.
- 2d/dx (x³) = 3x²
- 3d/dx (−6x²) = −12x
- 4d/dx (9x) = 9
- 5f′(x) = 3x² − 12x + 9
Answer: f′(x) = 3x² − 12x + 9
Exam tips
- Always simplify before differentiating — never differentiate a product if you can expand it first.
- For sketching cubics: find intercepts, stationary points, then check end-behaviour.
- Optimisation: read the question twice, define your variable, write the constraint, then differentiate.
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