Introduction to Analysis · Chapter 07

Differentiation

The derivative as a limit, and what one theorem about a flat point gives: the Mean Value Theorem, monotonicity, the counterexamples that show why each hypothesis is there, and Taylor with its remainder.

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

(f g)^{'} = f^{'} g + f g^{'}

(f \circ g)^{'} = (f^{'} \circ g) g^{'}

f^{''} (x)

\frac{f ( b ) - f ( a )}{b - a}

f^{(n)} (a)

R_{n} (x)

\frac{d}{d x} e^{x} = e^{x}

Lessons

Build the foundation

01The derivative as a limitThe average rate of change of a function over a stretch is the quotient of the change in the values by the change in the inputs. Shrink the stretch and ask for the limit: if it exists, it is the derivative. The quotient is undefined at the point itself, which is the whole reason the last chapter insisted that a limit never looks at the value at the point.Read lesson →
02Differentiability and continuityA function with a derivative at a point is continuous there, and the proof is three lines. The converse is false, and it fails badly: there are functions continuous everywhere with a derivative nowhere. The gap between the two conditions is the first place in this course where a picture is actively misleading, because every curve a hand can draw has a tangent almost everywhere and the general continuous function does not.Read lesson →
03Rules of differentiationThe rules for sums, products and quotients are not conventions to memorise; each is a short argument with the difference quotient, and the argument is worth seeing once because it explains the shape of the answer. The product rule looks strange until the trick is seen: add and subtract a middle term, so that one factor is held still while the other moves.Read lesson →
04The chain ruleRates multiply along a composition. If the inner function changes three times as fast as its input and the outer one changes twice as fast as its input, the composition changes six times as fast — and the outer rate must be measured where the inner function actually lands, not where the input is. That is the whole rule, and the only difficulty in proving it is that the inner step may be zero.

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05Interior extrema and Rolle’s theoremAt an interior point where a differentiable function is largest, the derivative must be zero: a non-zero rate would let the values be increased by stepping one way or the other. That one observation, combined with the Extreme Value Theorem, gives the first of the two great theorems of this chapter: a differentiable function with equal values at the two ends of an interval is flat somewhere inside.Read lesson →

06The Mean Value TheoremOver any interval a differentiable function has, at some interior point, an instantaneous rate equal to its average rate over the whole interval. It is Rolle with the interval tilted: subtract the chord and the two end values become equal again, so Rolle applies to the difference and its flat point is the point where the original function runs parallel to the chord.Read lesson →

07What the Mean Value Theorem givesEverything a derivative tells you about a function on an interval comes from this one theorem. A zero derivative means constant; a positive derivative means increasing; two functions with the same derivative differ by a constant; and a bounded derivative bounds how fast the values can move. None of these is obvious from the definition, because the definition is about one point at a time, and all of them are about a whole interval.Read lesson →

08Derivatives that are not continuousA derivative need not be continuous. That is worth saying slowly, because every derivative met so far has been, and because the natural guess — that differentiating a differentiable function gives a continuous one — is false. What is true is stranger: a derivative cannot have a jump, because it always has the intermediate value property, whether or not it is continuous. So the ways a derivative can misbehave are limited, and none of them is a jump.Read lesson →

09Taylor's theoremThe tangent line is the best linear approximation to a function at a point, and its error vanishes faster than the step. Using more derivatives gives a polynomial whose error vanishes faster still, and the theorem says exactly how much faster: the error is the next term of the same shape, with the derivative evaluated at an unknown point inside the interval. It is the Mean Value Theorem with more derivatives, and its proof is Rolle applied to a cleverly chosen function.Read lesson →

10Chapter practiceThis lesson mixes the chapter without saying which idea each question is about. The tools are the definition, the four rules with the chain rule, and the single theorem from which everything else in the chapter follows.Read lesson →

Mapping the lesson.

Differentiation

Build the foundation