Differentiability

The derivative of f at c is f′(c), the limit of [f(c + h) − f(c)]/h as h → 0, provided it exists. Geometrically it is the slope of the tangent: the limit of the slopes of chords through (c, f(c)).

f is differentiable at c when the left-hand derivative (h → 0⁻) and the right-hand derivative (h → 0⁺) are both finite and equal.

Rules from Class XI: (u ± v)′ = u′ ± v′, (uv)′ = u′v + uv′, and (u/v)′ = (u′v − uv′)/v² where v ≠ 0. Standard results: (xⁿ)′ = nxⁿ⁻¹, (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec²x.

Theorem: if f is differentiable at c, it is continuous at c. Proof idea: f(x) − f(c) equals the chord slope times (x − c), which tends to f′(c) × 0 = 0.

So every differentiable function is continuous. The converse fails: |x| is continuous at 0, but its left-hand derivative there is −1 and its right-hand derivative is +1, so it is not differentiable at 0.

Corners and jumps both spoil differentiability. |x − 1| is not differentiable at 1, and [x] is not differentiable at 1 or 2 (it is not even continuous there).

A function is differentiable on [a, b] when it is differentiable at every point, using the one-sided derivative at each end.