Differentiation from first principles
KS5 · 40 minutes · Derive and apply the derivative.
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Our learning goal
Derive and apply the derivative.
Your turn: Choose one thing you would like to understand by the end.
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Full teaching script
- Today: Differentiation from first principles. Our goal: Derive and apply the derivative.
- Take one idea at a time. Pause after each sentence; ask for repetition whenever you need it.
- The gradient of a curve at a point is the limit of the chord gradient as h→0: f′(x) = lim (f(x+h) − f(x))/h. For xⁿ that yields nxⁿ⁻¹. Derivatives give rates of change, stationary points and optimisation.
- Together: choose one example from the explanation and show how it works. What do you notice?
- Your turn: Differentiate from first principles: f(x) = x², then f(x) = 3x² − x.
- Reflect: what made sense, and what would you like explained differently?
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