Theory

Differential calculus and Taylor

The derivative as the limit of the difference quotient, the differentiation rules, the theorems of Fermat, Rolle and Lagrange with their consequences, L'Hôpital's rule and Taylor's formula.

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The derivative
Differentiability and continuity

The derivative

1. Derivative and difference quotient

Let $f : A \to R$ and let $x_{0}$ be an interior point of $A$ . The difference quotient of $f$ at $x_{0}$ relative to the increment $h \neq = 0$ is

\frac{f ( x _{0} + h ) - f ( x _{0} )}{h},

the slope of the line through the points $(x_{0}, f (x_{0}))$ and $(x_{0} + h, f (x_{0} + h))$ of the graph. $f$ is differentiable at $x_{0}$ if the following limit exists and is finite:

f^{'} (x_{0}) = h \to 0 lim \frac{f ( x _{0} + h ) - f ( x _{0} )}{h} = x \to x_{0} lim \frac{f ( x ) - f ( x _{0} )}{x - x _{0}};

the two expressions coincide under the substitution $x = x_{0} + h$ . If $f$ is differentiable at every point of a set, the function $x \mapsto f^{'} (x)$ is the derivative function. The derivative measures how quickly $f$ changes near $x_{0}$ ; its sign is used to study monotonicity, maxima and minima, as the theorems of this chapter show.

y = x^{2}

y = 3 x - 2

For

f (x) = x^{2}

x_{0} = 1

and

h = 1

the difference quotient equals

\frac{4 - 1}{1} = 3

: it is the slope of the secant

y = 3 x - 2

(dashed)

Derivative of x² using the definition

h \to 0 lim \frac{( x + h ) ^{2} - x ^{2}}{h} = h \to 0 lim \frac{2 x h + h ^{2}}{h} = h \to 0 lim (2 x + h) = 2 x

The limit exists and is finite for every $x \in R$ , therefore $f (x) = x^{2}$ is differentiable on $R$ and $f^{'} (x) = 2 x$ .

2. One-sided derivatives

The right derivative and the left derivative of $f$ at $x_{0}$ are the one-sided limits of the difference quotient:

f_{+}^{'} (x_{0}) = h \to 0^{+} lim \frac{f ( x _{0} + h ) - f ( x _{0} )}{h}, f_{-}^{'} (x_{0}) = h \to 0^{-} lim \frac{f ( x _{0} + h ) - f ( x _{0} )}{h} .

$f$ is differentiable at $x_{0}$ if and only if the two one-sided derivatives exist, are finite and coincide. They are used at the junction points of piecewise-defined functions and at the points where an absolute value changes expression.

3. Tangent line to the graph

If $f$ is differentiable at $x_{0}$ , the tangent line to the graph of $f$ at the point $(x_{0}, f (x_{0}))$ is

y = f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) .

As $h \to 0$ the point $(x_{0} + h, f (x_{0} + h))$ approaches $(x_{0}, f (x_{0}))$ and the secant line through the two points tends to the tangent; since the slope of the secant is the difference quotient, the slope of the tangent is $f^{'} (x_{0})$ . This is the geometric meaning of the derivative.

y = x^{2}

y = 3 x - 2

y = 2 x - 1

The secants through

(1, 1)

with

h = 1

and with

h = \frac{1}{2}

(dashed) have slopes

3

and

\frac{5}{2}

; as

h \to 0

they tend to the tangent

y = 2 x - 1

(red), with slope

f^{'} (1) = 2

y = x^2

y = 2 x - 1

y = x^{2}

: at the point

(1, 1)

the tangent line (dashed) is

y = 2 x - 1

, since

f^{'} (1) = 2

Differentiability and continuity

1. Differentiable implies continuous

If $f$ is differentiable at $x_{0}$ , then $f$ is continuous at $x_{0}$ . This is why the study of differentiability begins with continuity: at a point where $f$ is not continuous, $f$ is not differentiable.

Proof

For $x \neq = x_{0}$ the following identity holds:

f (x) - f (x_{0}) = \frac{f ( x ) - f ( x _{0} )}{x - x _{0}} \cdot (x - x_{0}) .

As $x \to x_{0}$ the first factor tends to $f^{'} (x_{0})$ and the second to $0$ , therefore by the limit of a product $f (x) - f (x_{0}) \to f^{'} (x_{0}) \cdot 0 = 0$ , that is, $f (x) \to f (x_{0})$ , which is the definition of continuity at $x_{0}$ .

Warning

The implication does not reverse: $f (x) = ∣ x ∣$ is continuous at $0$ but is not differentiable at $0$ . The difference quotient $\frac{∣ h ∣}{h}$ equals $1$ for $h > 0$ and $- 1$ for $h < 0$ , therefore the one-sided derivatives exist and are different.

2. Corner points, cusps, vertical-tangent inflection points

Let $f$ be continuous at $x_{0}$ but not differentiable at $x_{0}$ . The point is classified by the one-sided limits of the difference quotient.

Corner point: the one-sided derivatives exist, are different and at least one is finite. Example: $f (x) = ∣ x ∣$ at $0$ , with one-sided derivatives $- 1$ and $1$ .
Cusp: the difference quotient tends to $+ \infty$ on one side and to $- \infty$ on the other. Example: $f (x) = ∣ x ∣$ at $0$ : the quotient equals $\frac{1}{h}$ for $h > 0$ and $- \frac{1}{- h}$ for $h < 0$ .
Inflection point with vertical tangent: the difference quotient tends to $+ \infty$ or to $- \infty$ on both sides. Example: $f (x) = 3 x$ at $0$ , where the quotient $\frac{3 h}{h} = \frac{1}{3 h ^{2}}$ tends to $+ \infty$ ; the tangent is the vertical line $x = x_{0}$ .

y = \left|x\right|

y = ∣ x ∣

: corner point at

0

y = \sqrt{\left|x\right|}

y = ∣ x ∣

: cusp at

0

y = 3 x

: inflection point with vertical tangent at

0

The theory continues with 6 more sections (differentiation rules, maxima minima and mean value theorems, consequences of lagranges theorem…), plus all worked examples, the solutions and the PDF.

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