Derivatives as Linear Approximation

f is differentiable at a with f'(a)=m if the error E(h)=f(a+h)−f(a)−mh satisfies lim_{h→0} E(h)/h = 0 — i.e. the affine model L(x)=f(a)+m(x−a) has error shrinking faster than |x−a|.

Any function whose ratio to h tends to 0 as h→0; 'negligible compared to h itself'. Differentiability says f(a+h)=f(a)+f'(a)h+o(h).

f(a+h) = f(a) + f'(a)h + h·ε(h), where ε(h)→0 as h→0. It's the main algebraic tool for proving the differentiation rules.

Let h→0 in f(a+h)=f(a)+f'(a)h+hε(h): both the f'(a)h and hε(h) terms vanish, so f(a+h)→f(a). No ε–δ argument needed.

Continuity only says f(a+h)=f(a)+o(1) (error small). Differentiability says that after subtracting a linear term, the error is o(h) — small *relative to h*.

No single slope m works on both sides: the best linear model has slope −1 from the left and +1 from the right, so lim E(h)/h doesn't exist.

L(x)=10+(1/20)(x−100), so √101 ≈ 10.05 (true 10.049875…, error ≈1.25×10⁻⁴).

The error is not merely o(h) but O(h²), the quadratic behavior Taylor's theorem guarantees for sufficiently nice functions.

Add the two expansions: (f+g)(a+h)=(f+g)(a)+(f'(a)+g'(a))h+h(ε₁+ε₂), and ε₁+ε₂→0. Hence (f+g)'(a)=f'(a)+g'(a).

It extends verbatim to multivariable calculus (derivative = best linear map), differential geometry, and underlies Newton's method and gradient descent: linearize, step, re-linearize.