10. 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 the other way around. This scene zooms into the corner that shows why. Continuous only means the graph has no breaks. Differentiable asks for more. The tangent slope must settle to a single value. At a sharp corner the slope splits into two. For example, a 🚗 car climbing a smooth hill has a speed at every moment. But on a path with a sharp zigzag corner, the speed at the corner is not one single value.
• A Discontinuous Derivative — A function differentiable everywhere whose derivative alone is discontinuous. Zoom near the origin to watch the derivative oscillate. The function itself settles smoothly at the origin. Its slope, though, never stops oscillating. Differentiable does not force the derivative to be continuous. Still, the derivative never skips an intermediate value. For example, picture a ⚽ ball bouncing lower and faster each time it hits the floor. Its height calmly sinks to zero. Its bounce speed, though, keeps flickering to the very end.
• The Secant Meets Its Parallel Tangent — Somewhere on the curve sits a tangent parallel to the secant. This scene lets you hunt for that point. The secant's slope is the average rate of change. At some interior point the instantaneous slope matches it exactly. That is what the mean value theorem guarantees. For example, a 🚗 car covering 100 km in one hour averages 100 km/h. At some moment in that hour, the speedometer read exactly 100.
• Derivative Sign and Monotonicity — The sign of the derivative decides where the curve rises and falls. Drag the point and watch the tangent's color track the sign. Where the slope is positive, the curve climbs. Where it is negative, the curve drops. Reading the sign alone reveals the monotone stretches. The mean value theorem proves this link. For example, while you climb a 🪜 staircase your height keeps growing. While you go down, it keeps shrinking. The tilt alone tells you whether you rise or fall.
• The Inverse Function Theorem — Near a point with nonzero slope, a function can be flipped. The flipped function's slope is the reciprocal of the original. Flipping swaps horizontal and vertical. So the ratio of rise to run turns upside down. Multiply the two tangent slopes and you get exactly 1. For example, imagine 💰 money and 🍬 candies traded at a 🏪 shop: 4 money for 1 candy. Traded back, 1 money buys a quarter of a candy — the rate flips into its reciprocal.
• Cauchy's Mean Value Theorem — The mean value theorem survives on parametric curves, where two functions move together. A tangent parallel to the chord always exists. Track the change of one quantity against another. Somewhere, the instantaneous ratio equals the whole interval's ratio. This is the bridge that proves l'Hôpital's rule. For example, a 🚗 car climbing a mountain road gains distance and height together. At some moment it climbs at exactly the trip's overall height-to-distance ratio.
• L'Hôpital's Rule — This scene tackles limits that look like 0 divided by 0. Differentiate top and bottom separately and the limit appears. When both sides shrink to zero, amounts stop deciding the ratio. What matters is which one shrinks faster. Comparing the two speeds gives the answer. For example, when two puddles of 💧 water both dry up, their remaining ratio is not set by the starting amounts. It is set by the ratio of their drying speeds.
• Darboux's Theorem — A derivative keeps the intermediate value property even when discontinuous. Watch a wildly oscillating derivative step on every level. A derivative cannot jump across values. To get from one value to another it must pass through everything between. Even a discontinuous derivative obeys this rule. For example, a 🌡️ thermometer sliding from −1° to 1° passes through every temperature between. A derivative likewise never jumps over an intermediate value.
• The Weierstrass Function — A function continuous everywhere yet differentiable nowhere. Add partial sums and the graph grows thornier. Ordinary curves flatten when you zoom in. This one grows fresh teeth at every magnification. So no point ever earns a tangent line. Uniform convergence powers the continuity half. For example, zoom into a 🌳 tree branch and smaller twigs appear, then smaller still, forever. At no magnification does the jaggedness smooth away.
• Polynomials Hugging the Curve — This scene lays a polynomial over a curve. Raise the degree and it hugs the curve. Match the value, the slope, and the bending at one point. The more you match, the better the imitation nearby. Far away, the imitation eventually breaks. For example, knowing where a 🚗 car is, how fast it goes, and how it accelerates lets you predict its position a little later. More information predicts farther ahead.
• Taylor's Theorem: The Remainder Envelope — This scene shows the band that traps the Taylor error. Raise the degree and the band shrinks to zero. The remainder has a ceiling. So you can promise in advance how large the error can get. The true curve always stays inside the band. For example, measuring with a 📏 ruler promises an error of at most a millimeter. The Taylor remainder makes the same kind of promise. Higher degree narrows the promised range.
• Convex Functions — This scene studies functions whose chords float above the graph. Convexity turns out to be the derivative increasing. Picture a bowl shape opening upward. A curve whose slope keeps growing is convex. The middle height never exceeds the average of the end heights. For example, stretch a 🧵 string taut between two points on a bowl-shaped curve and it always floats above the curve. The string's middle sits higher than the curve's middle.
• Riding Tangents to the Root — Ride tangent lines down to a root. Each iteration doubles the number of correct digits. Draw the tangent and take its root as the next guess. A straight line stands in for the curve. From a good start, the guesses converge astonishingly fast. For example, guessing a number a 🧑 friend hid, each hint doubles the correct digits of your next guess. A wild first guess turns exact within a few rounds.
• The Contraction Cobweb — A cobweb spirals into a single point. That point is the fixed point. Repeatedly apply a map that shrinks distances by a fixed ratio. The points crowd closer and closer together. In the end only one point can remain. For example, nest a smaller 📦 box inside a box, then a smaller one inside that, and the boxes close in on a single point. Each step leaves less room than the last.