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The derivative from first principles
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Learn it The derivative is the instantaneous rate of change: f′(x) = lim (h→0) [f(x + h) − f(x)]/h, the slope of the tangent. The average rate over smaller and smaller intervals approaches it.
Example: f(x) = x². Find f′(3) from first principles.
- [(3 + h)² − 9]/h = (6h + h²)/h = 6 + h.
- As h → 0: f′(3) = 6.
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