Introducing the Derivative 3.1

3.1: Introducing the Derivative

Slope of a Line

The slope of a straight line measures its steepness and is defined as the ratio of the vertical change to the horizontal change between two points on the line.

• Formula: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$

• Interpretation: 'rise over run'—the change in $y$ divided by the change in $x$.

Example: For points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is calculated as above.

Slope of the Secant Line and Tangent Line

For a curve $y = f(x)$, the secant line passes through two points on the curve, while the tangent line touches the curve at a single point and has the same instantaneous direction as the curve at that point.

• Secant Line Slope: $m_{sec} = \dfrac{f(x) - f(a)}{x - a}$

• Tangent Line Slope: $m_{tan} = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$

The tangent line's slope is the limit of the secant line's slope as the two points approach each other.

Examples: Finding the Slope of the Tangent Line

• Example 1: Let $f(x) = 3x + 4$ at $P(2, 10)$

  • Find $m_{tan}$: $m_{tan} = \lim_{x \to 2} \dfrac{f(x) - f(2)}{x - 2}$ $= \lim_{x \to 2} \dfrac{3x + 4 - 10}{x - 2}$ $= \lim_{x \to 2} \dfrac{3x - 6}{x - 2}$ $= \lim_{x \to 2} 3 = 3$

• Example 2: Let $f(x) = x^2 + 6x$ at $P(1, 7)$

  • Find $m_{tan}$: $m_{tan} = \lim_{x \to 1} \dfrac{x^2 + 6x - 7}{x - 1}$ $= \lim_{x \to 1} \dfrac{(x + 7)(x - 1)}{x - 1}$ $= \lim_{x \to 1} (x + 7) = 8$

  • Equation of tangent line: $y - 7 = 8(x - 1) \implies y = 8x - 1$

Alternative Definition: Slope of the Tangent Line

The slope of the tangent line can also be defined using an increment $h$:

• Secant Line Slope: $m_{sec} = \dfrac{f(a + h) - f(a)}{h}$

• Tangent Line Slope: $m_{tan} = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$

This form is often used to define the derivative at a point.

Examples: Tangent Line Using the $h$-Definition

• Example 3: $f(x) = -2x + 7$ at $(2, 3)$

  • $m_{tan} = \lim_{h \to 0} \dfrac{f(2 + h) - f(2)}{h}$

  • $= \lim_{h \to 0} \dfrac{-2(2 + h) + 7 - 3}{h}$

  • $= \lim_{h \to 0} \dfrac{-4 - 2h + 4}{h}$

  • $= \lim_{h \to 0} \dfrac{-2h}{h} = -2$

  • Equation: $y - 3 = -2(x - 2) \implies y = -2x + 7$

• Example 4: $f(x) = x^2 + 4x$ at $(1, 5)$

  • $m_{tan} = \lim_{h \to 0} \dfrac{(1 + h)^2 + 4(1 + h) - 5}{h}$

  • $= \lim_{h \to 0} \dfrac{h^2 + 6h}{h}$

  • $= \lim_{h \to 0} (h + 6) = 6$

  • Equation: $y - 5 = 6(x - 1) \implies y = 6x - 1$

Definition of the Derivative at a Point

The derivative of a function $f(x)$ at a point $a$ is defined as the limit of the average rate of change as the interval shrinks to zero.

• Definition 1: $f'(a) = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$

• Definition 2: $f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$

The derivative represents the instantaneous rate of change of $f$ at $a$, or the slope of the tangent line at $(a, f(a))$.

Example: Computing the Derivative at a Point

• Example 5: $f(x) = 2x^2 - 3x$, compute $f'(3)$

  • $f'(3) = \lim_{h \to 0} \dfrac{f(3 + h) - f(3)}{h}$

  • $= \lim_{h \to 0} \dfrac{2(3 + h)^2 - 3(3 + h) - [2 \cdot 3^2 - 3 \cdot 3]}{h}$

  • $= \lim_{h \to 0} \dfrac{2(9 + 6h + h^2) - 9 - 3h - (18 - 9)}{h}$

  • $= \lim_{h \to 0} \dfrac{18 + 12h + 2h^2 - 9 - 3h - 9}{h}$

  • $= \lim_{h \to 0} \dfrac{12h + 2h^2 - 3h}{h}$

  • $= \lim_{h \to 0} \dfrac{9h + 2h^2}{h}$

  • $= \lim_{h \to 0} (9 + 2h) = 9$

Summary Table: Definitions of the Derivative

Form	Expression	Interpretation
Difference Quotient (x-form)	$f'(a) = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$	Limit of average rate of change as $x \to a$
Difference Quotient (h-form)	$f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$	Limit of average rate of change as $h \to 0$

Key Points

• The tangent line to a curve at a point has a slope equal to the derivative at that point.

• The derivative gives the instantaneous rate of change of a function.

• Both the $x$-form and $h$-form of the difference quotient are used to define the derivative.

• Finding the equation of a tangent line involves computing the derivative and using the point-slope form of a line.