Definition of the Derivative

Limit Definition

The derivative of $f$ at $x$ is defined as:

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Alternative form:

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Notation

The derivative can be written in several ways:

Notation	Read as
$f'(x)$	"f prime of x"
$\frac{dy}{dx}$	"dy dx" or "derivative of y with respect to x"
$\frac{d}{dx}[f(x)]$	"d dx of f of x"
$Df(x)$	"D of f of x"
$\dot{y}$	"y dot" (Newton's notation)

Geometric Interpretation

The derivative $f'(a)$ represents:

• The slope of the tangent line to $y = f(x)$ at $x = a$
• The instantaneous rate of change of $f$ at $x = a$

Tangent Line Equation

The tangent line to $y = f(x)$ at point $(a, f(a))$:

$y - f(a) = f'(a)(x - a)$

or

$y = f(a) + f'(a)(x - a)$

Physical Interpretation

If $s(t)$ represents position at time $t$:

• $s'(t) = v(t)$ = velocity
• $s''(t) = v'(t) = a(t)$ = acceleration

Differentiability

A function $f$ is differentiable at $x = a$ if $f'(a)$ exists.

Conditions for Differentiability

• $f$ must be continuous at $a$ (necessary but not sufficient)
• The left and right derivatives must exist and be equal

Non-Differentiable Points

A function is NOT differentiable at:

• Corners (sharp points)
• Cusps
• Vertical tangents
• Discontinuities

Examples

Example 1: Using the definition

Find $f'(x)$ for $f(x) = x^2$:

$f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h}$

$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$

$= \lim_{h \to 0} \frac{2xh + h^2}{h}$

$= \lim_{h \to 0} (2x + h) = 2x$

Example 2: Tangent line

Find the tangent line to $f(x) = x^3$ at $x = 2$:

• $f(2) = 8$
• $f'(x) = 3x^2$, so $f'(2) = 12$

Tangent line: $y - 8 = 12(x - 2)$, or $y = 12x - 16$