Derivatives: synthesis and challenges

Summary

This lesson closes the unit on derivatives. It gathers into a single deductive chain what the ten lessons of the unit proved: the derivative as the limit of the difference quotient, its three equivalent characterizations and the differentiation rules, all obtained from the Carathéodory factorization without dividing by quantities that may vanish, together with the derivatives of the elementary functions and the chain rule; the intermediate value theorem and the continuity of the inverse, which support the derivative of the inverse function, and the reduction of powers and exponentials to any base to the natural exponential; and higher-order derivatives, the differential and rates of change. It explains how the unit rests on the preceding unit and on Algebra and Geometry I and II, and how it prepares the applications of derivatives. It ends with an exit assessment, parallel to the entry assessment, and with ten challenges, harder than the problems of the lessons, each of which combines results from several of them.

Learning objectives

Set out in logical order the main definitions and theorems of the unit, indicating which earlier results each one depends on and in which proofs the Carathéodory factorization is reused.
Distinguish what the unit admits from what it proves, and recognize the hypotheses of each result, such as that of the derivative of the inverse or the one required by implicit differentiation.
Relate the results of the unit to the unit on limits and continuity, to Algebra and Geometry I and II and to the applications of derivatives.
Solve non-routine problems that combine the definition of the derivative, the rules, the chain rule, the intermediate value theorem, the derivative of the inverse, powers with a variable exponent, higher-order derivatives, the differential and rates of change.

What the unit achieved

The whole unit rests on one idea: the derivative is a limit, and the preceding unit had already provided everything needed to compute it and handle it. But the difference quotient is a division, and dividing by an increment that may vanish is the source of defective proofs. The Carathéodory characterization replaces the division by a product, with a factor $\varphi$ continuous at $a$ :

f(x) - f(a) = \varphi(x)(x - a), \qquad \varphi(a) = f'(a)

With it, proving that a function is differentiable reduces to factoring its increment with a continuous factor, and the continuity of the factor is decided by the theorems of the preceding unit. This is how the rules for linear combinations, products and reciprocals, the chain rule and the derivative of the inverse function were obtained, each one citing the earlier ones instead of redoing them.

The definition and the rules

The lesson on the derivative as a limit reached the same definition from the tangent and from velocity, and settled where it makes sense: at a point of the domain that is an accumulation point of it. It wrote the derivative as a quantified statement, defined the one-sided derivatives as one-sided limits of the difference quotient, the tangent and normal lines, the derivative function and Leibniz notation, and proved that the derivative admits three equivalent formulations: the limit of the difference quotient, the approximation $f(a + h) = f(a) + f'(a)h + \operatorname{o}(h)$ and the Carathéodory factorization. From the last one it followed in two lines that differentiable implies continuous; the absolute value and the cube root showed that the converse fails. The lesson on differentiation rules first proved that the derivative depends only on what happens near the point, and then, by combining factorizations, the rules for linear combinations, products, reciprocals and quotients; with them it differentiated powers with an integer exponent, polynomials and rational functions, and decided differentiability at the junction point of two pieces, where the derivatives of the formulas are not enough. The lesson on derivatives of the elementary functions obtained those of sine and cosine from the fundamental trigonometric limit and the sum formulas, those of the natural exponential and of $\ln \abs{x}$ from the standard limits, without the chain rule or the inverse function rule, and those of the other trigonometric functions and of the hyperbolic functions with the rules. The lesson on the chain rule combined two factorizations, explained why the proof that divides by $g(x) - g(a)$ fails when the inner function repeats its value, took care of the domain of the composition and wrote the rule in Leibniz notation, where it looks like a cancellation of fractions without being one.

Continuity, inverses and exponents

The lesson on the intermediate value theorem proved Bolzano’s theorem with the supremum of the points where the function is negative and the preservation of sign, and from it obtained the intermediate value theorem, the fact that the image of an interval under a continuous function is an interval and the fact that the inverse of a continuous, strictly monotonic function on an interval is continuous; the bisection method turned the proof into a procedure. The lesson on the derivative of the inverse function read the Carathéodory factorization backward: if $f'(a) \neq 0$ , the factor does not vanish, and its reciprocal, composed with the continuous inverse, factors the increment of the inverse; if $f'(a) = 0$ , the chain rule prevents the inverse from being differentiable. With that theorem it differentiated the inverse trigonometric functions, the inverse hyperbolic functions, roots and powers with a rational exponent, and it differentiated implicitly defined functions, assuming in each case that they exist and are differentiable. The lesson on the variable in the exponent reduced everything to the natural exponential, with $a^x = e^{x \ln a}$ and $\left(f(x)\right)^{g(x)} = e^{g(x) \ln f(x)}$ : it differentiated the exponential and the logarithm to any base, powers with a real exponent and those with variable base and exponent, justified logarithmic differentiation by proving differentiability before taking logarithms, and decided that $\pi^e < e^\pi$ with the tangent to the logarithm that passes through the origin.

Higher-order derivatives and rates of change

The lesson on higher-order derivatives and the differential defined the derivatives of higher order by recursion, with their domain, and proved by induction the formulas of order $n$ for powers, the exponential, sine and cosine, and the Leibniz rule, which has the coefficients of the binomial theorem. It gave separate meaning to $\mathrm{d}y$ and $\mathrm{d}x$ : the differential $\mathrm{d}y = f'(x) \, \mathrm{d}x$ is the only linear part of the increment that differs from it by a quantity negligible compared with $\Delta x$ ; it used it to approximate values and to estimate the propagation of errors, without bounding the error of the approximation, which requires later tools. The lesson on rates of change read the derivative as how fast a quantity changes: it distinguished the average, instantaneous and relative rates, characterized growth at a constant relative rate, described motion along a line with velocity, speed and acceleration, proved what the sign of the derivative at a point says about nearby values, and no more, and founded on the chain rule an explicit method for related rates.

What is admitted, what is proved and what remains

The balance of the unit can be summarized as follows:

Admitted are the completeness axiom, which in this unit only Bolzano’s theorem uses directly, the theorems of the unit on limits and continuity and results proved in Algebra and Geometry: the sum formulas, the fundamental identities, the laws of exponents and of logarithms, the inequality $e^x \geq 1 + x$ , the monotonicity of the functions that define the inverses, induction and the binomial theorem. It is also assumed, in each case, that an equation defines a differentiable function when it is differentiated implicitly.
Proved are facts that Algebra and Geometry had admitted or announced: the intermediate value theorem, the fact that the tangent to the graph of $e^x$ at $(0, 1)$ has slope $1$ and that $e$ is the only base whose exponential has that slope; and the tangent, which for conics was defined as the line that touches the curve without crossing it, comes to be defined for any graph.
Four tools are reused instead of repeating arguments: the Carathéodory factorization, the theorem that differentiable implies continuous, the local character of the derivative and the preservation of sign.
Left for later is what requires more than the derivative at a point: the mean value theorem and what follows from it, monotonicity from the sign of the derivative, extrema, L’Hôpital’s rule and the error bounds for the differential, in the following unit; the Taylor polynomial and the integral, in Calculus II; and the theorem that guarantees implicitly defined functions, in multivariable calculus.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that come from the preceding unit or from Algebra and Geometry, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

ACompleteness axiom. It is admitted, from Algebra and Geometry I, that the reals form a complete ordered field: every nonempty set of reals that is bounded above has a supremum. Admitted without proof in this unit.
BInheritance from unit 1. Results proved in the unit on limits and continuity: the algebra of limits, with the limit of a restriction and that of a composition; one-sided limits; continuity, that of the operations and that of composition; the continuity of the elementary functions and of the inverse of a monotonic function whose image is an interval; the limits of $\frac{\sin h}{h}$ , $\frac{1 - \cos h}{h}$ , $\frac{e^h - 1}{h}$ and $\frac{\ln(1 + h)}{h}$ at $0$ ; and little $\operatorname{o}$ and equivalence. Admitted without proof in this unit.
CPreservation of sign. Proved in the unit on limits and continuity: if $f$ tends at $a$ to a limit $L \neq 0$ , near $a$ the function has the sign of $L$ ; and if $f$ is continuous at $a$ with $f(a) \neq 0$ , near $a$ it has the sign of $f(a)$ and does not vanish. Admitted without proof in this unit.
DInheritance from Algebra and Geometry. Results proved in Algebra and Geometry I and II: induction, the binomial theorem and Pascal’s rule; the sum formulas and the fundamental trigonometric and hyperbolic identities; the laws of exponents and of logarithms, change of base and the inequality $e^x \geq 1 + x$ ; and the monotonicity of the restrictions that define the inverse trigonometric functions and of the hyperbolic functions. Admitted without proof in this unit.
ECarathéodory factorization. $f$ is differentiable at $a$ , with $f'(a) = L$ , if and only if there exists $\varphi$ , continuous at $a$ , with $\varphi(a) = L$ and $f(x) - f(a) = \varphi(x)(x - a)$ on the whole domain: to differentiate it suffices to factor the increment, without dividing. Established in: 3. Three characterizations.
FDifferentiable implies continuous. Every function differentiable at a point is continuous there; the rules use this so that the Carathéodory factor turns out continuous. Established in: 4. Differentiable implies continuous.
GLocal character. The derivative at $a$ does not change when the function is restricted to a set of which $a$ is still an accumulation point, and it depends only on the values near $a$ ; that is why one works on the intersection of the domains or on an open interval. Established in: 5. Local character of the derivative.

Segment 1 The definition and the rules
- Define the derivative at an accumulation point of the domain as the limit of the difference quotient, with its one-sided derivatives, and write it as a quantified statement.
- Prove the equivalence of the difference quotient, the linear approximation and the Carathéodory factorization, and deduce that differentiable implies continuous, but not conversely.
- Obtain with factorizations, without dividing by increments, the rules for linear combinations, products, reciprocals and quotients, and differentiate polynomials, rational functions and piecewise functions.
- Differentiate the trigonometric functions, the exponential, the logarithm and the hyperbolic functions with the limits of the preceding unit, and their compositions with the chain rule.
1. Definition of the derivative. If $a$ is an element of the domain $A$ of $f$ that is an accumulation point of $A$ , then $f'(a) = L$ when $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - a} < \delta \rightarrow \abs{\frac{f(x) - f(a)}{x - a} - L} < \varepsilon\right)$ ; equivalently, $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L$ .
  Rests on: B (Inheritance from unit 1). Proved in: The derivative as the limit of a function.
2. One-sided derivatives. The one-sided derivatives $f'_+(a)$ and $f'_-(a)$ are the one-sided limits of the difference quotient; if $a$ is an accumulation point of the domain from both sides, $f$ is differentiable at $a$ if and only if both exist and are equal, and if it is one from only one side, the derivative is the one from that side.
  Uses: 1. Definition of the derivative. Rests on: B (Inheritance from unit 1). Proved in: The derivative as the limit of a function.
3. Three characterizations. The following are equivalent: $f'(a) = L$ ; $f(a + h) = f(a) + Lh + \operatorname{o}(h)$ as $h$ tends to $0$ ; and the existence of a function $\varphi$ , continuous at $a$ , with $\varphi(a) = L$ , such that $f(x) - f(a) = \varphi(x)(x - a)$ for every $x$ in the domain.
  Uses: 1. Definition of the derivative. Rests on: B (Inheritance from unit 1). Proved in: The derivative as the limit of a function.
4. Differentiable implies continuous. If $f$ is differentiable at $a$ , it is continuous at $a$ ; the converse is false: $\abs{x}$ and $\sqrt[3]{x}$ are continuous at $0$ and are not differentiable at $0$ .
  Uses: 2. One-sided derivatives, 3. Three characterizations. Rests on: B (Inheritance from unit 1), E (Carathéodory factorization). Proved in: The derivative as the limit of a function.
5. Local character of the derivative. If $F$ is the restriction of $f$ to a set $B$ that contains $a$ and of which $a$ is an accumulation point, and $f$ is differentiable at $a$ , then $F'(a) = f'(a)$ ; and if $B$ contains all the points of the domain near $a$ , $f$ is differentiable at $a$ if and only if $F$ is.
  Uses: 1. Definition of the derivative. Rests on: B (Inheritance from unit 1). Proved in: Differentiation rules.
6. Linear combination and product. If $f$ and $g$ are differentiable at $a$ , so are $\alpha f + g$ and $fg$ , with $(\alpha f + g)'(a) = \alpha f'(a) + g'(a)$ and $(fg)'(a) = f'(a) \, g(a) + f(a) \, g'(a)$ .
  Uses: 3. Three characterizations, 4. Differentiable implies continuous. Rests on: B (Inheritance from unit 1), E (Carathéodory factorization), F (Differentiable implies continuous). Proved in: Differentiation rules.
7. Reciprocal and quotient. If $f$ and $g$ are differentiable at $a$ and $g(a) \neq 0$ , then $a$ is an accumulation point of the domain of $\frac{f}{g}$ , and $\left(\frac{1}{g}\right)'(a) = -\frac{g'(a)}{\left(g(a)\right)^2}$ and $\left(\frac{f}{g}\right)'(a) = \frac{f'(a) \, g(a) - f(a) \, g'(a)}{\left(g(a)\right)^2}$ hold.
  Uses: 3. Three characterizations, 4. Differentiable implies continuous, 5. Local character of the derivative, 6. Linear combination and product. Rests on: B (Inheritance from unit 1), C (Preservation of sign), E (Carathéodory factorization), F (Differentiable implies continuous), G (Local character). Proved in: Differentiation rules.
8. Powers, polynomials and rational functions. For every integer $m \neq 0$ , the derivative of $x^m$ is $m x^{m - 1}$ on its whole domain; every polynomial is differentiable on $\mathbb{R}$ , with the term-by-term derivative, and every rational function is differentiable on its domain.
  Uses: 1. Definition of the derivative, 6. Linear combination and product, 7. Reciprocal and quotient. Rests on: D (Inheritance from Algebra and Geometry). Proved in: Differentiation rules.
9. Derivative at a junction point. If $f$ coincides with $g$ to the left of $c$ and with $h$ to the right, with $g$ and $h$ differentiable at $c$ , then $f$ is differentiable at $c$ if and only if $g(c) = f(c) = h(c)$ and $g'(c) = h'(c)$ ; comparing the derivatives of the formulas is not enough.
  Uses: 2. One-sided derivatives, 4. Differentiable implies continuous. Rests on: B (Inheritance from unit 1), F (Differentiable implies continuous). Proved in: Differentiation rules.
10. Trigonometric functions. For every real $x$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin x\right) = \cos x$ and $\frac{\mathrm{d}}{\mathrm{d}x}\left(\cos x\right) = -\sin x$ ; where $\cos x \neq 0$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\tan x\right) = \frac{1}{\cos^2 x} = 1 + \tan^2 x$ ; and the cotangent, the secant and the cosecant are differentiated with the quotient and reciprocal rules.
  Uses: 1. Definition of the derivative, 6. Linear combination and product, 7. Reciprocal and quotient. Rests on: B (Inheritance from unit 1), D (Inheritance from Algebra and Geometry). Proved in: Derivatives of the elementary functions.
11. Exponential, logarithm and hyperbolic functions. $\frac{\mathrm{d}}{\mathrm{d}x}\left(e^x\right) = e^x$ on $\mathbb{R}$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\ln \abs{x}\right) = \frac{1}{x}$ on $\mathbb{R} \setminus \{0\}$ , and the derivatives of $\sinh x$ , $\cosh x$ and $\tanh x$ are $\cosh x$ , $\sinh x$ and $\frac{1}{\cosh^2 x}$ .
  Uses: 1. Definition of the derivative, 6. Linear combination and product, 7. Reciprocal and quotient. Rests on: B (Inheritance from unit 1), D (Inheritance from Algebra and Geometry). Proved in: Derivatives of the elementary functions.
12. Chain rule. If $g$ is differentiable at $a$ and $f$ , defined on a set that contains the values of $g$ , is differentiable at $g(a)$ , then $(f \circ g)'(a) = f'(g(a)) \, g'(a)$ ; the proof that divides by $g(x) - g(a)$ is valid only when $g'(a) \neq 0$ .
  Uses: 3. Three characterizations, 4. Differentiable implies continuous. Rests on: B (Inheritance from unit 1), C (Preservation of sign), E (Carathéodory factorization), F (Differentiable implies continuous). Proved in: The chain rule for the derivative of a composition of functions.
13. Standard compositions. The rule holds on the domain of $f \circ g$ when $a$ is an accumulation point of it, which happens if the domain of $f$ contains an interval centered at $g(a)$ ; with it, $\frac{\mathrm{d}}{\mathrm{d}x}\left(\left(g(x)\right)^m\right) = m \left(g(x)\right)^{m - 1} g'(x)$ for every integer $m \neq 0$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sqrt{g(x)}\right) = \frac{g'(x)}{2\sqrt{g(x)}}$ if $g(a) > 0$ , and $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin g(x)\right) = \cos g(x) \cdot g'(x)$ .
  Uses: 1. Definition of the derivative, 5. Local character of the derivative, 8. Powers, polynomials and rational functions, 10. Trigonometric functions, 12. Chain rule. Rests on: C (Preservation of sign), F (Differentiable implies continuous), G (Local character). Proved in: The chain rule for the derivative of a composition of functions.
Segment 2 Continuity, inverses and exponents
- Prove Bolzano’s theorem with the supremum and, with it, the intermediate value theorem, the fact that the image of an interval under a continuous function is an interval and the fact that the inverse of a continuous, strictly monotonic function is continuous.
- Prove the derivative of the inverse function by reading the Carathéodory factorization backward, and differentiate the inverse trigonometric functions, the inverse hyperbolic functions, roots and powers with a rational exponent.
- Differentiate implicitly defined functions, distinguishing the computation of the derivative from the proof that it exists.
- Reduce to the natural exponential the exponentials and logarithms to any base and the powers with a real exponent or with variable base and exponent, and justify logarithmic differentiation.
1. Bolzano’s theorem. If $f$ is continuous on $\icc{a}{b}$ and $f(a) < 0 < f(b)$ , or $f(a) > 0 > f(b)$ , there exists $c$ in $\ioo{a}{b}$ such that $f(c) = 0$ .
  Rests on: A (Completeness axiom), C (Preservation of sign). Proved in: The intermediate value theorem.
2. Intermediate value theorem. If $f$ is continuous on $\icc{a}{b}$ , it takes every value between $f(a)$ and $f(b)$ ; therefore, the image of an interval under a continuous function is an interval.
  Uses: 14. Bolzano’s theorem. Rests on: B (Inheritance from unit 1). Proved in: The intermediate value theorem.
3. Continuity of the inverse. If $f$ is continuous and strictly monotonic on an interval $I$ , its inverse is continuous on the interval $f(I)$ .
  Uses: 15. Intermediate value theorem. Rests on: B (Inheritance from unit 1). Proved in: The intermediate value theorem.
4. Derivative of the inverse function. If $f$ is continuous and strictly monotonic on an interval $I$ with more than one point and differentiable at $a$ , with $f'(a) \neq 0$ , then $f^{-1}$ is differentiable at $b = f(a)$ , and $\left(f^{-1}\right)'(b) = \frac{1}{f'(a)}$ ; if $f'(a) = 0$ , $f^{-1}$ is not differentiable at $b$ .
  Uses: 1. Definition of the derivative, 3. Three characterizations, 12. Chain rule, 15. Intermediate value theorem, 16. Continuity of the inverse. Rests on: B (Inheritance from unit 1), E (Carathéodory factorization). Proved in: Derivative of the inverse function and implicit differentiation.
5. Elementary inverses and roots. On their domains of differentiability, $\frac{\mathrm{d}}{\mathrm{d}x}\left(\arcsin x\right) = \frac{1}{\sqrt{1 - x^2}}$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\arctan x\right) = \frac{1}{1 + x^2}$ and $\frac{\mathrm{d}}{\mathrm{d}x}\left(\operatorname{arcsinh} x\right) = \frac{1}{\sqrt{x^2 + 1}}$ ; $\sqrt[n]{x}$ is differentiable at every $x \neq 0$ of its domain and not at $0$ ; and $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^{\frac{p}{q}}\right) = \frac{p}{q} x^{\frac{p}{q} - 1}$ on $\ioo{0}{\infty}$ .
  Uses: 8. Powers, polynomials and rational functions, 10. Trigonometric functions, 11. Exponential, logarithm and hyperbolic functions, 13. Standard compositions, 17. Derivative of the inverse function. Rests on: B (Inheritance from unit 1), D (Inheritance from Algebra and Geometry). Proved in: Derivative of the inverse function and implicit differentiation.
6. Exponential and logarithm to base a. If $a > 0$ and $a \neq 1$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(a^x\right) = a^x \ln a$ and $\frac{\mathrm{d}}{\mathrm{d}x}\left(\log_a \abs{x}\right) = \frac{1}{x \ln a}$ ; and if $u$ is differentiable at a point, so is $e^{u(x)}$ , with derivative $e^{u(x)} u'(x)$ , and if moreover $u$ does not vanish there, so is $\ln \abs{u(x)}$ , with derivative $\frac{u'(x)}{u(x)}$ .
  Uses: 6. Linear combination and product, 11. Exponential, logarithm and hyperbolic functions, 12. Chain rule, 13. Standard compositions. Rests on: D (Inheritance from Algebra and Geometry). Proved in: The variable in the exponent and logarithmic differentiation.
7. Powers with real and variable exponents. For every real $r$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^r\right) = r x^{r - 1}$ on $\ioo{0}{\infty}$ ; and if $f$ , with positive values, and $g$ are differentiable at $a$ , then $F(x) = \left(f(x)\right)^{g(x)} = e^{g(x) \ln f(x)}$ is differentiable at $a$ , with $F'(a) = \left(f(a)\right)^{g(a)} \left(g'(a) \ln f(a) + \frac{g(a) f'(a)}{f(a)}\right)$ .
  Uses: 6. Linear combination and product, 11. Exponential, logarithm and hyperbolic functions, 12. Chain rule, 19. Exponential and logarithm to base a. Rests on: D (Inheritance from Algebra and Geometry). Proved in: The variable in the exponent and logarithmic differentiation.
8. Logarithmic differentiation. If $f_1, \ldots, f_k$ are differentiable at $a$ and do not vanish at $a$ , and $f_j$ is positive when $r_j$ is not an integer, then $F = f_1^{r_1} \cdots f_k^{r_k}$ is differentiable at $a$ , and $\frac{F'(a)}{F(a)} = r_1 \frac{f_1'(a)}{f_1(a)} + \cdots + r_k \frac{f_k'(a)}{f_k(a)}$ ; differentiability is proved before taking logarithms.
  Uses: 4. Differentiable implies continuous, 5. Local character of the derivative, 6. Linear combination and product, 13. Standard compositions, 19. Exponential and logarithm to base a, 20. Powers with real and variable exponents. Rests on: D (Inheritance from Algebra and Geometry), F (Differentiable implies continuous), G (Local character). Proved in: The variable in the exponent and logarithmic differentiation.
9. Pi to the e and e to the pi. For every $x > 0$ , $x^{\frac{1}{x}} \leq e^{\frac{1}{e}}$ holds, with equality only if $x = e$ , because the logarithm lies below its tangent $y = \frac{x}{e}$ ; in particular, $\pi^e < e^\pi$ .
  Uses: 11. Exponential, logarithm and hyperbolic functions. Rests on: D (Inheritance from Algebra and Geometry). Proved in: The variable in the exponent and logarithmic differentiation.
Segment 3 Higher-order derivatives and rates of change
- Define higher-order derivatives and prove by induction the formulas of order $n$ for powers, the exponential, sine and cosine, and the Leibniz rule.
- Give separate meaning to $\mathrm{d}y$ and $\mathrm{d}x$ , prove that the differential is the principal part of the increment and use it to approximate and to estimate errors, without bounding them.
- Read the derivative as an average, instantaneous and relative rate of change, and describe with it motion along a line.
- Solve related rates problems with the chain rule and an explicit method.
1. Higher-order derivatives. $f^{(0)} = f$ and $f^{(n + 1)} = \left(f^{(n)}\right)'$ , with the domain of each one formed by the points at which the preceding one is differentiable; if $f$ and $g$ are $n$ times differentiable on an open interval, $\left(\alpha f + g\right)^{(n)} = \alpha f^{(n)} + g^{(n)}$ .
  Uses: 6. Linear combination and product. Rests on: D (Inheritance from Algebra and Geometry), G (Local character). Proved in: Higher-order derivatives and the differential.
2. Derivatives of order n. On $\ioo{0}{\infty}$ , $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(x^r\right) = r(r - 1)\cdots(r - n + 1) \, x^{r - n}$ ; for every real $x$ , $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(e^{cx}\right) = c^n e^{cx}$ and $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\sin x\right) = \sin\left(x + \frac{n\pi}{2}\right)$ ; and the coefficients of a polynomial $f$ are the numbers $\frac{f^{(k)}(0)}{k!}$ .
  Uses: 8. Powers, polynomials and rational functions, 13. Standard compositions, 19. Exponential and logarithm to base a, 20. Powers with real and variable exponents, 23. Higher-order derivatives. Rests on: D (Inheritance from Algebra and Geometry). Proved in: Higher-order derivatives and the differential.
3. Leibniz rule. If $f$ and $g$ are $n$ times differentiable on an open interval, so is $fg$ , and $(fg)^{(n)} = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)} g^{(n - k)}$ .
  Uses: 6. Linear combination and product, 23. Higher-order derivatives. Rests on: D (Inheritance from Algebra and Geometry). Proved in: Higher-order derivatives and the differential.
4. The differential. If $f$ is differentiable at $a$ , the increment $\Delta y = f(a + \Delta x) - f(a)$ satisfies $\Delta y = \mathrm{d}y + \operatorname{o}(\Delta x)$ , with $\mathrm{d}y = f'(a) \, \Delta x$ , and no other linear function of $\Delta x$ satisfies this; if $f'(a) \neq 0$ , $\Delta y \sim \mathrm{d}y$ ; and if $w = f(u)$ , then $\mathrm{d}w = f'(u) \, \mathrm{d}u$ .
  Uses: 3. Three characterizations, 6. Linear combination and product, 7. Reciprocal and quotient, 12. Chain rule. Rests on: B (Inheritance from unit 1). Proved in: Higher-order derivatives and the differential.
5. Constant relative rate. If $y(t) = y_0 e^{kt}$ , with $y_0 > 0$ and $k \neq 0$ , the relative rate of change $\frac{y'(t)}{y(t)}$ equals $k$ at every instant, and $y(t + T) = e^{kT} y(t)$ : the quantity doubles, or is halved, in each time interval of length $\frac{\ln 2}{\abs{k}}$ .
  Uses: 19. Exponential and logarithm to base a. Rests on: D (Inheritance from Algebra and Geometry). Proved in: Rates of change and related rates.
6. The sign of the derivative at a point. If $f$ is defined on an open interval $I$ , is differentiable at a point $a$ of $I$ and $f'(a) > 0$ , there exists $\delta > 0$ such that, for every $x$ in $I$ , $f(x) < f(a)$ if $a - \delta < x < a$ , and $f(x) > f(a)$ if $a < x < a + \delta$ ; it does not follow that $f$ is increasing near $a$ . Hence the speed increases when the velocity and the acceleration have the same sign.
  Uses: 1. Definition of the derivative, 6. Linear combination and product. Rests on: C (Preservation of sign). Proved in: Rates of change and related rates.
7. Related rates. If $u$ is differentiable at $t_0$ and $y(t) = g(u(t))$ , with $g$ differentiable at $u(t_0)$ , then $y'(t_0) = g'(u(t_0)) \, u'(t_0)$ ; and two expressions that are equal at every instant of an interval and differentiable at $t_0$ have the same derivative at $t_0$ .
  Uses: 12. Chain rule. Proved in: Rates of change and related rates.

Toward the applications. The chain stops where the derivative ceases to be the object of study and becomes a tool: the following unit begins with Weierstrass’s theorem on extreme values; it proves Fermat’s theorem with the sign of the derivative at a point, and from it Rolle’s theorem and the mean value theorem, which turn the sign of the derivative into monotonicity and yield L’Hôpital’s rule, extrema, concavity and the bounds that the differential lacks; and it ends with the Newton-Raphson method, which replaces the curve by its tangent.