Pure · Differentiation
Chapter 1 · 4
The idea
Differentiation from first principles
The definition every differentiation rule is built on — the gradient as the limit of a chord — and the "cancel the h, then let h → 0" technique for using it directly.
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Pure · Differentiation
Differentiation from first principles
The definition every differentiation rule is built on — the gradient as the limit of a chord — and the "cancel the h, then let h → 0" technique for using it directly.
Why it works
The chord that becomes a tangent
Every differentiation rule you'll ever use rests on a single definition, and "from first principles" means going back to it instead of using a shortcut. The exam asks this directly — "differentiate from first principles" is a regular 3–4 marker, and quoting from the power rule scores none of it: the marks are for the definition and the limit, shown step by step.Sneaking up on a curve's gradient
The gradient of a straight line is easy — rise over run. A curve's gradient keeps changing, so we sneak up on it. Take the point at and a second point a small step further along, at . The straight line joining them (a chord) has gradient Now slide that second point in, letting shrink toward : the chord pivots until it becomes the tangent, and its gradient becomes the gradient of the curve itself. That limit is the derivative:Keep reading — free
The rest of the explanation, plus 2 worked examples you step through move by move.
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