Linear Approximation and Differentials

This section builds the chapter's two central tools for a function of one variable: the linear approximation (the tangent-line formula that lets us estimate $f$ near a chosen point without computing $f$ exactly) and the differential (a precise algebraic object $df$ that captures the change in $f$ predicted by that tangent line). Along the way we make precise what it means to say an approximation is "off by a little" — the absolute, relative, and percentage error — since every real approximation needs an honest measure of how good it is.

The unifying starting point is the definition of the derivative itself: for $f:(a,b)\to\mathbb R$ differentiable at $x\in(a,b)$,

When $\Delta x$ is small (but not zero), this says $\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\approx f'(x)$, i.e.

$f(x+\Delta x)-f(x) \approx f'(x)\,\Delta x, \qquad\text{i.e.}\qquad f(x+\Delta x)\approx f(x)+f'(x)\Delta x.$ …