Theory

Derivatives, marginal functions and elasticity

What the derivative is and how to compute it with the differentiation rules, what marginal cost, marginal revenue, marginal profit and the elasticity of demand measure, and what Fermat's, Rolle's and Lagrange's theorems and de l'Hôpital's rule say.

On this page

Difference quotient and derivative
Derivatives of elementary functions

Difference quotient and derivative

1. Derivative at a point

Let $f$ be defined on an open interval containing $x_{0}$ . The difference quotient of $f$ between $x_{0}$ and $x_{0} + h$ , with $h \neq = 0$ , is

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

that is, the change in $f$ divided by the change in the variable; on the graph it is the slope of the secant line through the points with abscissas $x_{0}$ and $x_{0} + h$ . The function is differentiable at $x_{0}$ if the limit

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

exists and is finite; it is called the derivative of $f$ at $x_{0}$ and is also denoted by $\frac{df}{d x} (x_{0})$ . For $f (x) = x^{2}$ and $x_{0} = 3$ the difference quotient is $\frac{( 3 + h ) ^{2} - 9}{h} = \frac{6 h + h ^{2}}{h} = 6 + h$ , which tends to $6$ ; therefore $f^{'} (3) = 6$ . The function $x \mapsto f^{'} (x)$ , defined where $f$ is differentiable, is called the derivative function.

2. Tangent line

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

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

As $h$ tends to $0$ the second point of the secant approaches the first, and the secant tends to the line through $(x_{0}, f (x_{0}))$ with slope $f^{'} (x_{0})$ ; the formula is that of the line through a point with a given slope. For $f (x) = x^{2}$ at $x_{0} = 3$ the tangent is $y = 9 + 6 (x - 3) = 6 x - 9$ .

y = x^2

y = 6x - 9

y = 5x - 6

The secant through the points with abscissas

2

and

3

has slope

5

; as the second point approaches

3

, the secant tends to the tangent

y = 6 x - 9

, with slope

f^{'} (3) = 6

3. Differentiability and continuity

If $f$ is differentiable at $x_{0}$ , then $f$ is continuous at $x_{0}$ .

The converse is false. For example $f (x) = ∣ x ∣$ is continuous at $0$ , but the difference quotient $\frac{∣ h ∣}{h}$ equals $1$ for $h > 0$ and $- 1$ for $h < 0$ , so it has no limit and the function is not differentiable at $0$ . On the graph the point $(0, 0)$ is a corner point, where the slopes from the right and from the left are different.

y = \left|x\right|

The graph of

∣ x ∣

has slope

- 1

to the left of

0

and slope

1

to the right, so at

(0, 0)

it has a corner point and the function is not differentiable at

0

Proof

Write the difference $f (x_{0} + h) - f (x_{0})$ as the product of the difference quotient and $h$ :

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

because the first factor tends to the finite number $f^{'} (x_{0})$ and the second to $0$ . Therefore $f (x_{0} + h) \to f (x_{0})$ , that is, $f$ is continuous at $x_{0}$ .

Derivatives of elementary functions

At the points of their domain where they are differentiable, the following hold

$f (x)$	$f^{'} (x)$
$k$ (constant)	$0$
$x^{a}$ , with $a$ real	$a x^{a - 1}$
$e^{x}$	$e^{x}$
$a^{x}$ , with $a > 0$	$a^{x} ln a$
$ln x$	$\frac{1}{x}$
$sin x$	$cos x$
$cos x$	$- sin x$

The formulas are derived from the definition using the standard limits. For the exponential

\frac{e ^{x + h} - e ^{x}}{h} = e^{x} \cdot \frac{e ^{h} - 1}{h} \to e^{x},

and for the logarithm, with $x > 0$ ,

\frac{ln ( x + h ) - ln x}{h} = \frac{1}{x} \cdot \frac{ln ( 1 + \frac{h}{x} )}{\frac{h}{x}} \to \frac{1}{x} .

The power formula includes roots and reciprocals: $(x)^{'} = (x^{\frac{1}{2}})^{'} = \frac{1}{2 x}$ and $(\frac{1}{x})^{'} = (x^{- 1})^{'} = - \frac{1}{x ^{2}}$ . The function $x$ is not differentiable at $0$ , where $\frac{1}{2 x}$ is not defined.

The theory continues with 6 more sections (differentiation rules, marginal functions, elasticity…), plus all worked examples, the solutions and the PDF.

Unlock now