9. Differentiation

From the definition of the derivative through the mean value theorem, l'Hôpital, and Taylor expansions.

• Differentiable Implies Continuous — Differentiable implies continuous, but not conversely — zoom into the corner of $|x|$ and it never flattens. The first step of the differentiation scenes.
• The Secant Meets Its Parallel Tangent — Somewhere on the curve, a tangent is parallel to the secant of slope $\frac{f(b)-f(a)}{b-a}$. Drag $\xi$ to find it.
• Cauchy's Mean Value Theorem — On the parametric curve $(g(t), f(t))$ a tangent parallel to the chord exists. For $f(t) = t^2$, $g(t) = t^3$ it sits at $\xi = 14/9$.
• L'Hôpital's Rule — In a $0/0$ or $\infty/\infty$ form, differentiating the numerator and the denominator keeps the limit. Drive $x$ toward the limit point and watch $\frac{f(x)}{g(x)}$ track $\frac{f'(x)}{g'(x)}$.
• Darboux's Theorem — Derivatives have the intermediate value property even when discontinuous — $f'(x) = 2x\sin(1/x) - \cos(1/x)$ oscillates wildly near 0 yet steps on every level. Jump discontinuities are impossible in a derivative.
• The Weierstrass Function — $W(x) = \sum_{n=0}^{\infty} a^n\cos(b^n\pi x)$ — continuous everywhere yet differentiable nowhere. The M-test scene powers the continuity argument, and the graph turns thornier as partial sums grow. The full nowhere-differentiability proof is idea-level only.
• Polynomials Hugging the Curve — Lay the Maclaurin polynomial $T_n(x)$ over $\sin x$. Raise the degree and watch it hug the curve.
• Taylor's Theorem: The Remainder Envelope — The error $R_n(x)$ is trapped in the band $T_n(x) \pm |x|^{n+1}/(n+1)!$. Raise $n$ and the band shrinks to zero.
• Convex Functions — Convex when the chord floats above the graph — for differentiable $f$ this is the same as $f'$ increasing, and the second derivative decides it. Up to Jensen's inequality.
• Riding Tangents to the Root — Ride the tangent down to the root. The number of correct digits doubles each step.
• The Contraction Cobweb — The cobweb between $y = x$ and $y = f(x)$ gets sucked into one point. The error shrinks like $q^n$.