Pure · Differentiation
Chapter 1 · 4
The idea
Differentiating powers of x
Bring the power down to the front and knock it down by one — and why that rule is exactly what the gradient-of-a-chord limit always produces.
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Pure · Differentiation
Differentiating powers of x
Bring the power down to the front and knock it down by one — and why that rule is exactly what the gradient-of-a-chord limit always produces.
Why it works
What gradient even means here
Differentiation has one job: find the gradient of a curve — how steep it is — at a point. The catch is that "gradient" was only ever defined for straight lines (rise over run). A curve's steepness changes everywhere, so what does its gradient at a single point even mean?Sneak up with a chord
The trick is to sneak up on it. Take two points on the curve and join them with a straight line — a chord — whose gradient you can find, it's just rise over run. Now slide the second point towards the first. The chord pivots, and its gradient closes in on the steepness of the curve at that one point: the tangent. 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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