The derivative as a linear approximation
f(a+h) ≈ f(a) + f′(a)h
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What this is really saying
Intuition
Close to a point, a differentiable function behaves almost like its tangent line. The derivative supplies the slope of that local linear model.
Example
If f(x)=x² near x=3, then f′(3)=6, so a small change h gives f(3+h)≈9+6h.