Course 2 Book 2A: Numbers, Limits and the IntegralChapter 10

Derivatives

The mean value theorem, the second-derivative test, and the first finite-time blow-up.

30 min read · Updated Oct 2, 2026

Read with Tao, Analysis I, chapter "Differentiation of functions" (basic definitions, local maxima and minima, monotone functions and derivatives, inverse functions and derivatives, L'Hôpital's rule).

In this chapter · 8 sections

10.1The derivative
10.1.1The rules
10.2Local extrema
10.3The mean value theorem
10.4Inverse functions
10.5L'Hôpital's rule
10.6The derivative of a maximum
10.7Comparison and blow-up
10.8Exercises

Course 1 used derivatives as a tool. This chapter founds them. The definition is a limit, and the theorems follow from the completeness of $\mathbb{R}$ through the maximum principle of 2A.9 Continuous Functions. On the way we meet the facts that this guidebook leans on most heavily:

the derivative is the best linear approximation, the idea that generalises to every dimension (2B.8 Calculus in Several Variables) and to curved spaces (8A.3 Tangent Vectors and Bundles);
at an interior maximum, $f' = 0$ and $f'' \leq 0$ , the seed of every maximum principle;
the mean value theorem, which turns information about derivatives into information about values;
comparison for simple differential inequalities, and the first finite-time blow-up. The chapter ends by showing, with nothing more than this, that Ricci flow on a positively curved space must develop a singularity in finite time, and that the round 3-sphere shrinks to a point at exactly the time this argument predicts.

By the end of this chapter you will be able to:

prove differentiability and the rules of differentiation from the definition;
prove and use Rolle's theorem and the mean value theorem;
explain why $f' = 0$ and $f'' \leq 0$ at an interior maximum;
differentiate a maximum of functions, where it has corners, using one-sided derivatives;
solve $y' = cy^2$ and show that a differential inequality $y' \geq cy^2$ forces blow-up in finite time.

The derivative

Definition 10.1 Derivative

Let $f$ be defined on an interval containing $x_0$ . Then $f$ is differentiable at $x_0$ with derivative $f'(x_0) = L$ if

\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = L.

The difference quotient is the slope of the chord between $(x_0, f(x_0))$ and $(x, f(x))$ , so the derivative is the limiting slope of chords, the slope of the tangent. An equivalent form says more.

Proposition 10.2 Newton's approximation

$f$ is differentiable at $x_0$ with derivative $L$ if and only if for every $\varepsilon > 0$ there is $\delta > 0$ such that

|f(x) - (f(x_0) + L(x - x_0))| \leq \varepsilon\,|x - x_0| \qquad\text{whenever } |x - x_0| < \delta.

This is just the definition multiplied through by $|x - x_0|$ , but it changes the point of view. The affine function $x \mapsto f(x_0) + L(x - x_0)$ approximates $f$ near $x_0$ with an error that is small compared with the distance $|x - x_0|$ . Any other line through $(x_0, f(x_0))$ has an error comparable to $|x - x_0|$ itself. So the tangent line is the unique best linear approximation. In several variables (2B.8 Calculus in Several Variables) this becomes the definition: the derivative is a linear map, not a number.

Figure 10.1. The tangent line at

x_0

(solid) and a chord (dashed). The error between

f

and its tangent, at distance

h

from

x_0

, shrinks faster than

h

(here like

h^2

). That is what makes the tangent the best linear approximation.

Differentiability implies continuity: $f(x) - f(x_0) = \frac{f(x) - f(x_0)}{x - x_0}(x - x_0) \to L \cdot 0 = 0$ . The converse fails. $|x|$ is continuous at $0$ but has chord slopes $-1$ from the left and $+1$ from the right, so it is not differentiable there. Stranger, $f(x) = x^2\sin(1/x)$ (with $f(0) = 0$ ) is differentiable at $0$ , with $f'(0) = 0$ , since $|f(x)/x| \leq |x|$ , yet $f'(x) = 2x\sin(1/x) - \cos(1/x)$ has no limit as $x \to 0$ . A function can be differentiable everywhere with a derivative that isn't continuous.

Figure 10.2. Left:

|x|

, continuous but not differentiable at

0

. Right:

x^2\sin(1/x)

, trapped between

\pm x^2

and so differentiable at

0

with derivative

0

, but with a derivative that oscillates wildly near

0

The rules

Theorem 10.3 Rules of differentiation

If $f$ and $g$ are differentiable at $x_0$ , then so are $f + g$ , $cf$ , $fg$ , and $f/g$ (if $g(x_0) \neq 0$ ), with

(fg)' = f'g + fg', \qquad \Big(\frac fg\Big)' = \frac{f'g - fg'}{g^2}.

Chain rule: if $f$ is differentiable at $x_0$ and $g$ is differentiable at $f(x_0)$ , then $g \circ f$ is differentiable at $x_0$ , with $(g \circ f)'(x_0) = g'(f(x_0))\,f'(x_0)$ .

Proof (The chain rule). The tempting proof writes $\frac{g(f(x)) - g(f(x_0))}{x - x_0} = \frac{g(f(x)) - g(f(x_0))}{f(x) - f(x_0)} \cdot \frac{f(x) - f(x_0)}{x - x_0}$ and takes limits. It fails when $f(x) = f(x_0)$ for $x$ arbitrarily close to $x_0$ , because then the first fraction divides by zero. Newton's approximation avoids this. Write $y_0 = f(x_0)$ and

g(y) = g(y_0) + g'(y_0)(y - y_0) + E(y)\,(y - y_0),

where $E(y) \to 0$ as $y \to y_0$ (and set $E(y_0) = 0$ ). Substituting $y = f(x)$ and dividing by $x - x_0$ :

\frac{g(f(x)) - g(f(x_0))}{x - x_0} = \big(g'(y_0) + E(f(x))\big)\,\frac{f(x) - f(x_0)}{x - x_0}.

As $x \to x_0$ , $f(x) \to y_0$ (by continuity), so $E(f(x)) \to 0$ , and the right side tends to $g'(y_0) f'(x_0)$ .

The lesson of this proof, to work with the error term rather than divide by something that might vanish, is the standard method for every chain rule later in the route, including the one on manifolds (8A.3 Tangent Vectors and Bundles).

Local extrema

Theorem 10.4 Derivatives vanish at interior extrema

If $f$ has a local maximum or minimum at an interior point $x_0$ of its domain, and $f$ is differentiable there, then $f'(x_0) = 0$ .

Proof. At a local maximum, $f(x) \leq f(x_0)$ for $x$ near $x_0$ . For $x > x_0$ the difference quotient is $\leq 0$ , so its limit $f'(x_0) \leq 0$ . For $x < x_0$ it is $\geq 0$ , so $f'(x_0) \geq 0$ . Hence $f'(x_0) = 0$ .

The proof needs $x_0$ to be interior, with room on both sides. At an endpoint only one inequality is available: on $[0, 1]$ , $f(x) = x$ has its maximum at $1$ , where $f'(1) = 1$ . Keeping track of boundary points is part of every maximum principle.

Theorem 10.5 The second derivative at a maximum

Suppose $f$ is differentiable near an interior point $x_0$ , twice differentiable at $x_0$ , and has a local maximum there. Then $f''(x_0) \leq 0$ .

Proof. We know $f'(x_0) = 0$ . Suppose $f''(x_0) > 0$ . Then $\frac{f'(x) - f'(x_0)}{x - x_0} = \frac{f'(x)}{x - x_0}$ is positive for $x$ close to $x_0$ , so $f'(x) > 0$ for $x$ slightly to the right of $x_0$ . By the mean value theorem below, $f$ is then strictly increasing on a small interval $[x_0, x_0 + \eta]$ , contradicting the maximum at $x_0$ .

So at an interior maximum the graph is flat and bends down. For a function of several variables the same is true in every direction, so the Hessian matrix is negative semidefinite, and in particular its trace, the Laplacian, is $\leq 0 $ (2B.8 Calculus in Several Variables). That is the inequality used in the maximum principle for the heat equation (6A.4 Maximum Principles) and, with the Laplacian of a Riemannian manifold, in Hamilton's maximum principle for Ricci flow (11A.4 Maximum Principles under Ricci Flow).

The mean value theorem

Theorem 10.6 Rolle's theorem

If $f$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f(a) = f(b)$ , then $f'(c) = 0$ for some $c \in (a, b)$ .

Proof. By the maximum principle (2A.9 Continuous Functions), $f$ attains a maximum and a minimum on $[a, b]$ . If both are attained only at the endpoints, then $f$ is constant (both equal $f(a) = f(b)$ ) and $f' = 0$ everywhere. Otherwise one of them is attained at an interior point $c$ , and $f'(c) = 0$ by Theorem 10.4.

Theorem 10.7 Mean value theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then

f'(c) = \frac{f(b) - f(a)}{b - a} \qquad\text{for some } c \in (a, b).

Proof. Apply Rolle to $g(x) = f(x) - \frac{f(b) - f(a)}{b - a}(x - a)$ , which has $g(a) = g(b) = f(a)$ .

Geometrically, some tangent is parallel to the chord. The chain of dependence is worth noticing: the mean value theorem uses Rolle, which uses the maximum principle, which uses Bolzano–Weierstrass, which uses the least upper bound property. Every theorem relating derivatives to values rests, ultimately, on the completeness of $\mathbb{R}$ .

In the world In use Average-speed cameras rely on the mean value theorem

On many roads, speed is enforced by section control: cameras at two points a known distance apart read each vehicle's number plate, and the system divides the distance by the time taken to get the average speed over the section. The UK's SPECS system and similar systems in several European countries work this way. The mean value theorem says that if a car's position $s(t)$ is differentiable, then at some instant between the two cameras its speed $s'(c)$ was exactly equal to the average speed. So an average above the limit proves the car was above the limit at some moment, even though no camera saw it there. A short burst of speed that is later compensated by slowing down may not raise the average enough to register, which is the price of measuring only the average.

Figure 10.3. Position against time between two cameras. The chord's slope is the average speed. The mean value theorem finds a time

c

at which the tangent is parallel to the chord: at that moment the car's speed equalled the average.

The mean value theorem converts derivative bounds into value bounds. Its most-used consequences:

Corollary 10.8 Consequences of the mean value theorem

Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$ .

If $f' > 0$ on $(a, b)$ , $f$ is strictly increasing; if $f' \geq 0$ , increasing; if $f' = 0$ , constant.
If $|f'| \leq K$ on $(a, b)$ , then $|f(x) - f(y)| \leq K|x - y|$ : $f$ is Lipschitz.
If $f' = g'$ on $(a, b)$ , then $f - g$ is constant.

Part 2 links derivatives to the uniform continuity of 2A.9 Continuous Functions. A bounded derivative gives a uniform modulus of continuity. In later courses, bounds on derivatives are exactly what make families of functions compact (2B.5 Uniform Convergence and Arzelà–Ascoli), and Shi's derivative estimates for Ricci flow (11A.3 Short-Time Existence and Uniqueness) are what make families of Ricci flows compact (11B.3 Compactness of Ricci Flows).

Inverse functions

Proposition 10.9 Derivative of an inverse

Let $f$ be continuous and strictly monotone on an interval, with inverse $f^{-1}$ . If $f$ is differentiable at $x_0$ with $f'(x_0) \neq 0$ , then $f^{-1}$ is differentiable at $y_0 = f(x_0)$ , with

(f^{-1})'(y_0) = \frac{1}{f'(x_0)}.

The proof is the definition, read through the continuous inverse of 2A.9 Continuous Functions. If $f'(x_0) = 0$ the inverse has a vertical tangent: $\sqrt[3]{y}$ , the inverse of $x^3$ , is not differentiable at $0$ .

In the world In use How sensitive is a thermometer?

In 2A.9 Continuous Functions, a digital thermometer computes temperature as $T = f^{-1}(R)$ from a measured resistance $R$ . Proposition 10.9 tells the designer how errors propagate: a small error $\Delta R$ produces a temperature error of about $\Delta R / f'(T)$ . Where the curve $R = f(T)$ is steep, small resistance errors barely matter; where it flattens, they are amplified. This is why each thermistor type has a stated useful range, the range over which $|f'(T)|$ is large enough for the target accuracy. In several variables the same question, how well an inverse is conditioned, is answered by the inverse function theorem (2B.9 The Inverse and Implicit Function Theorems), and in GPS positioning by the "dilution of precision".

L'Hôpital's rule

Proposition 10.10 L'Hôpital's rule

Let $f, g$ be differentiable near $x_0$ (except possibly at $x_0$ ), with $f(x), g(x) \to 0$ as $x \to x_0$ and $g'(x) \neq 0$ near $x_0$ . If $f'(x)/g'(x) \to L$ , then $f(x)/g(x) \to L$ .

It follows from a two-function version of the mean value theorem (Cauchy's) and is mainly a convenience. The examples worth remembering are the ones where it would mislead if misapplied. The hypothesis $f(x), g(x) \to 0$ is essential, and the rule says nothing when $f'/g'$ has no limit.

The derivative of a maximum

Suppose we track the largest of several quantities as time goes on: the hottest of several components, or the largest curvature on a manifold. The maximum $M(t) = \max_i F_i(t)$ of differentiable functions is usually not differentiable: it has corners where the leading function changes (Figure 10.4). Yet its rate of change is still controlled.

Definition 10.11 Upper right derivative

For a function $M$ , the upper right Dini derivative is $D^+M(t) = \limsup_{h \to 0^+} \frac{M(t + h) - M(t)}{h}$ , the steepest rate of increase seen immediately to the right of $t$ .

It always exists in $\mathbb{R}^*$ (2A.6 Sequences), and equals $M'(t)$ wherever $M$ is differentiable.

Proposition 10.12 Hamilton's trick, finite version

Let $F_1, \dots, F_k$ be differentiable, and $M(t) = \max_i F_i(t)$ . Then

D^+M(t) = \max\{F_i'(t) : i \text{ with } F_i(t) = M(t)\}.

In words: the maximum increases at the rate of the fastest-rising function among those currently attaining it.

Proof. Let $I$ be the set of indices with $F_i(t) = M(t)$ . For $i \notin I$ , $F_i(t) < M(t)$ , and by continuity $F_i < M$ on a short interval after $t$ , so these don't affect $M$ just after $t$ . For small $h > 0$ , then, $M(t + h) = \max_{i \in I} F_i(t + h)$ , and

\frac{M(t + h) - M(t)}{h} = \max_{i \in I} \frac{F_i(t + h) - F_i(t)}{h} \to \max_{i \in I} F_i'(t).

Figure 10.4. The maximum

M(t)

of three smooth functions (bold) has corners where the leader changes, so it isn't differentiable there. Its right-hand rate of increase is the slope of the fastest-rising function among those on top. That is Hamilton's trick.

In Ricci flow the "family" is the curvature at every point of a manifold, infinitely many functions, and the maximum over the manifold is attained because the manifold is compact. Hamilton's trick (11A.4 Maximum Principles under Ricci Flow) is the same statement: at any time, the maximum of the curvature changes at the rate $\partial_t F$ computed at a point where the maximum is attained. Combined with $\Delta F \leq 0$ at that point (Theorem 10.5, in several variables), this turns a PDE for $F$ into a differential inequality for $M(t)$ , which the next section shows how to solve.

Comparison and blow-up

A differential inequality such as $y' \geq g(y)$ doesn't determine $y$ , but it does bound it, by comparison with the solution of $z' = g(z)$ . The simplest example has dramatic consequences.

Example 10.13 Blow-up of

y' = c y^2

For $c > 0$ and $y(0) = y_0 > 0$ , the equation $y' = cy^2$ has the solution

y(t) = \frac{1}{1/y_0 - ct} = \frac{y_0}{1 - c\,y_0\,t}.

(Check: $y' = \frac{c\,y_0^2}{(1 - cy_0t)^2} = cy^2$ .) This tends to $+\infty$ as $t \to T = \frac{1}{c\,y_0}$ . The solution exists only for a finite time, and it blows up: the quantity becomes infinite in finite time. A linear equation $y' = cy$ can only grow exponentially, which never reaches infinity. Quadratic self-reinforcement can.

Proposition 10.14 Comparison

Let $y$ be differentiable on $[0, T)$ with $y(0) = y_0 > 0$ and $y' \geq cy^2$ , with $c > 0$ . Then $y(t) \geq \frac{y_0}{1 - c\,y_0\,t}$ for all $t$ in $[0, T)$ with $t < \frac{1}{c y_0}$ . In particular $T \leq \frac{1}{c y_0}$ : the solution can't survive past the blow-up time of the comparison equation.

Proof. Since $y' \geq 0$ , $y$ is increasing, so $y \geq y_0 > 0$ . Consider $u = 1/y$ . Then $u' = -y'/y^2 \leq -c$ , so by the mean value theorem $u(t) \leq u(0) - ct = 1/y_0 - ct$ . While the right side is positive this gives $y(t) \geq \frac{1}{1/y_0 - ct}$ . If the solution existed up to time $\frac{1}{cy_0}$ , then $u$ would have to become $\leq 0$ there, which is impossible since $u = 1/y > 0$ .

Figure 10.5. Solutions of

y' = y^2

starting at

y_0 = 1, 2, 4

blow up at

t = 1, \tfrac12, \tfrac14

: the larger the start, the sooner. The exponential solution of the linear equation

y' = y

(dashed) grows but stays finite for all time.

In the world Model Thermal runaway

When a chemical reaction releases heat, and its rate increases steeply with temperature (exponentially, by the Arrhenius law), a body that can't shed heat fast enough enters a feedback loop: hotter means faster means hotter still. The theory of thermal explosion developed by Nikolay Semenov and David Frank-Kamenetskii in the late 1920s and 1930s analysed exactly when heat loss can no longer keep up and the temperature runs away. The same feedback drives "thermal runaway" in lithium-ion batteries. The model $y' = cy^2$ is a deliberate simplification of these systems, not a model of any one of them. But it captures the mechanism: when the rate of growth grows faster than linearly with the quantity itself, the quantity can become infinite in finite time.

Where this goes A singularity in finite time, from first-year calculus

Here is the payoff, stated now and proved in 11A.4 Maximum Principles under Ricci Flow. Under Ricci flow on a closed $n$ -dimensional manifold, the scalar curvature $R$ evolves by $\partial_t R = \Delta R + 2|\mathrm{Ric}|^2$ , and $|\mathrm{Ric}|^2 \geq R^2/n$ (an inequality of linear algebra). At a point where $R$ is smallest, $\Delta R \geq 0$ , the minimum version of Theorem 10.5. Hamilton's trick (Proposition 10.12, for the minimum) then gives

\frac{d}{dt} R_{\min} \geq \frac{2}{n}\,R_{\min}^2.

If $R_{\min}(0) > 0$ , Proposition 10.14 with $c = 2/n$ says the flow must develop a singularity by time $T \leq \frac{n}{2R_{\min}(0)}$ .

Check it on the round unit 3-sphere ( $n = 3$ ). It has $R = n(n-1) = 6$ everywhere, so the bound is $T \leq \frac{3}{12} = \frac14$ . Under Ricci flow the sphere stays round while its radius shrinks, with radius squared $r(t)^2 = 1 - 4t$ , so $R(t) = 6/(1 - 4t)$ , which reaches infinity at exactly $t = \tfrac14$ . (Indeed $\frac{d}{dt}\frac{6}{1 - 4t} = \frac{24}{(1-4t)^2} = \frac23\big(\frac{6}{1 - 4t}\big)^2$ : the sphere satisfies the comparison equation with equality.) A singularity of the flow, predicted by a first-year comparison argument, and attained exactly by the most symmetric example. Perelman's work is largely about what happens at such singularities when the space is not so symmetric.

Recall Where we stand

The derivative is the best linear approximation. At an interior maximum it vanishes and the second derivative is non-positive. The mean value theorem turns derivative information into value information, and rests through Rolle and the maximum principle on the completeness of $\mathbb{R}$ . Maxima of families have one-sided derivatives given by Hamilton's trick, and differential inequalities like $y' \geq cy^2$ force blow-up in finite time. 2A.11 The Riemann Integral goes the other way, from rates of change back to totals: the integral.

Exercises

Exercise 10.15 From the definition

Prove from the definition that $f(x) = x^n$ has $f'(x) = nx^{n-1}$ for every natural number $n$ , by induction using the product rule. Then show that $\frac{d}{dx}x^{-n} = -nx^{-n-1}$ for $x \neq 0$ .

Exercise 10.16 Differentiable but not C¹

Show that $f(x) = x^2\sin(1/x)$ , $f(0) = 0$ , is differentiable everywhere, compute $f'$ , and show $f'$ is not continuous at $0$ . Does $f'$ have the intermediate value property anyway? (It does: Darboux's theorem says every derivative does. Try to prove it, using the maximum principle on $f(x) - yx$ .)

Exercise 10.17 Bounding values by derivatives

Use the mean value theorem to prove $|\sin x - \sin y| \leq |x - y|$ and $\ln(1 + x) \leq x$ for $x > -1$ (taking for granted the derivatives of $\sin$ and $\ln$ , constructed properly in 2B.6 Power Series, Exponentials and Bump Functions).

Exercise 10.18 Section control

Two cameras are $8$ km apart on a road with a $100$ km/h limit. A car passes them $4$ minutes $30$ seconds apart. (a) Compute its average speed, and explain what the mean value theorem allows you to conclude. (b) Could the car have exceeded $130$ km/h somewhere in the section? Could you prove it did?

Solution

(a) $8$ km in $4.5$ minutes is $8 / 0.075 \approx 106.7$ km/h. By the mean value theorem its speed was exactly $106.7$ km/h at some instant, so above the limit. (b) Possibly, yet nothing proves it: the data are consistent with a constant $106.7$ km/h, and also with bursts above $130$ balanced by slower stretches. Average speed bounds the maximum speed from below, but gives no upper bound.

Exercise 10.19 Hamilton's trick for the minimum

State and prove the version of Proposition 10.12 for $m(t) = \min_i F_i(t)$ , using the lower right Dini derivative $D_+ m(t) = \liminf_{h \to 0^+} \frac{m(t+h) - m(t)}{h}$ .

Exercise 10.20 Other powers

(a) Solve $y' = y^{1+\alpha}$ with $y(0) = y_0 > 0$ and $\alpha > 0$ , and find the blow-up time. (b) Show that $y' = y\,(1 + \ln y)$ (with $y_0 > 1$ ) does not blow up in finite time, though it grows faster than any exponential. (Substitute $u = \ln y$ .) Where exactly is the dividing line?

Solution

(a) $y^{-\alpha} = y_0^{-\alpha} - \alpha t$ , so $y = (y_0^{-\alpha} - \alpha t)^{-1/\alpha}$ , blowing up at $T = \frac{1}{\alpha y_0^{\alpha}}$ . (b) With $u = \ln y$ , $u' = 1 + u$ , so $u = (1 + u_0)e^t - 1$ , finite for all $t$ , and $y = \exp\big((1+u_0)e^t - 1\big)$ , a double exponential. The dividing line (Osgood's criterion) is whether $\int^\infty \frac{dy}{g(y)}$ is finite: it is for $g(y) = y^{1+\alpha}$ , and not for $g(y) = y\ln y$ .

Exercise 10.21 Rehearsal: the shrinking sphere

Under Ricci flow, a round $n$ -sphere of initial radius $r_0$ stays round with $r(t)^2 = r_0^2 - 2(n-1)t$ , and its scalar curvature is $R = n(n-1)/r^2$ . (a) Show that $R(t)$ satisfies $R' = \frac{2}{n}R^2$ exactly. (b) Find the extinction time, and check it agrees with the comparison bound $\frac{n}{2R(0)}$ . (c) A round 2-sphere of radius $1$ metre: when does it vanish, in the units in which the flow is $\partial_t g = -2\,\mathrm{Ric}$ ? This computation reappears as the first exact solution of Ricci flow in 11A.1 The Equation and Its First Solutions.

Solution

(a) $R = \frac{n(n-1)}{r_0^2 - 2(n-1)t}$ , so $R' = \frac{2n(n-1)^2}{(r_0^2 - 2(n-1)t)^2} = \frac{2}{n}\cdot\frac{n^2(n-1)^2}{(r_0^2 - 2(n-1)t)^2} = \frac2n R^2$ . (b) $r(t) = 0$ at $T = \frac{r_0^2}{2(n-1)}$ , and $\frac{n}{2R(0)} = \frac{n r_0^2}{2n(n-1)} = \frac{r_0^2}{2(n-1)}$ : equal. (c) $n = 2$ , $r_0 = 1$ : $T = \frac12$ (in square metres, the units of $r^2$ , since time in Ricci flow has the units of length squared).

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10.1The derivative

10.1.1The rules

10.2Local extrema

10.3The mean value theorem

10.4Inverse functions

10.5L'Hôpital's rule

10.6The derivative of a maximum

10.7Comparison and blow-up

10.8Exercises

The derivative

The rules

Local extrema

The mean value theorem

Inverse functions

L'Hôpital's rule

The derivative of a maximum

Comparison and blow-up

Exercises