Derivatives: Tangents, Rates of Change, and Basic Rules

Calculus and Analytic Geometry 1: Chapter 3 – Derivatives

3.1 Tangents and the Derivative at a Point

The concept of the derivative originates from the problem of finding the slope of the tangent line to the graph of a function at a specific point. This section introduces the foundational definition of the derivative and its geometric interpretation.

• Tangent Line: The tangent line to the curve y = f(x) at point P(x_0, f(x_0)) is the line through P with slope given by the limit:

$m = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• Secant Line: The secant line passes through two points on the curve and its slope is:

$\text{Secant slope} = \frac{f(z) - f(x)}{z - x}$

• Interpretation: As the second point approaches the first, the secant line approaches the tangent line.

Example 1

• Find the slope of the curve y = 1/x at any point x ≠ 0.

• Find the slope at x = -1.

• Where does the slope equal -1/2?

• What happens to the tangent as x changes?

Rates of Change: Derivative at a Point

The derivative at a point measures the instantaneous rate of change of a function at that point.

• Definition: The derivative of a function f at a point x_0, denoted f'(x_0), is:

$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• This limit must exist for the derivative to be defined.

Example 2

• Application: Find the exact speed of a rock falling freely from rest near the surface of the earth at a specific instant.

Example 3

• Find the slope of a given function's graph at a specified point and the equation of the tangent line.

Summary: Interpretations of the Derivative

The following are equivalent interpretations for the limit of the difference quotient:

$\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• The slope of the graph of y = f(x) at x = x_0

• The slope of the tangent to the curve y = f(x) at x = x_0

• The rate of change of f(x) with respect to x at x = x_0

• The derivative f'(x_0) at a point

3.2 The Derivative as a Function

The derivative can be viewed as a function itself, assigning to each value of x the slope of the tangent to the curve at that point.

• Definition: The derivative of the function f(x) with respect to x is the function f'(x) whose value at x is:

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

• Alternate form (using z instead of x + h):

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

• Differentiation: The process of calculating a derivative is called differentiation.

Example 1

• Differentiate f(x) = x/(x-1).

Example 2

• Find the derivative of f(x) = √x for x > 0.

• Find the tangent line to the curve y = √x at x = 4.

Graphing the Derivative

To graph the derivative of a function, estimate the slopes of the tangent lines at various points on the graph of f, then plot the points (x, f'(x)) in the xy-plane and connect them with a smooth curve.

• Procedure: Sketch the tangent to the graph at frequent intervals, estimate the slopes, and plot the corresponding pairs.

Differentiability on Intervals

A function y = f(x) is differentiable on an open interval if it has a derivative at each point of the interval. On a closed interval [a, b], differentiability requires the existence of one-sided limits at the endpoints.

Interval Type	Condition for Differentiability
Open interval (a, b)	Derivative exists at every point in (a, b)
Closed interval [a, b]	Derivative exists on (a, b) and one-sided limits exist at a and b

Example 4

• Show that f(x) = √x is differentiable on (0, ∞) and [0, ∞) but has no derivative at x = 0.

Example 5

• For x > 0, $\frac{d}{dx} \sqrt{x} = \frac{1}{2\sqrt{x}}$

• Examine the definition to see if the derivative exists at x = 0:

$\lim_{x \to 0^+} \frac{\sqrt{x} - 0}{x - 0} = \lim_{x \to 0^+} \frac{\sqrt{x}}{x} = \lim_{x \to 0^+} \frac{1}{\sqrt{x}}$

• Since the right-hand limit is not finite, there is no derivative at x = 0.

Where Does a Function Not Have a Derivative at a Point?

The derivative may fail to exist at a point for several reasons:

• Corner: One-sided derivatives differ.

• Cusp: Slope approaches infinity from one side and negative infinity from the other.

• Vertical Tangent: Slope approaches infinity from both sides.

• Discontinuity: The function is not continuous at the point.

• Oscillating Slope: The slope oscillates rapidly near the point (e.g., y = sin(1/x) near x = 0).

Theorem 1: Differentiability Implies Continuity

If f has a derivative at x = c, then f is continuous at x = c.

• Key Point: Differentiability is a stronger condition than continuity.

Power, Multiple, Sum, and Difference Rules

These rules simplify the process of finding derivatives for common types of functions.

• Derivative of a Constant Function: If f(x) = c, then:

$\frac{d}{dx} c = 0$

• Derivative of a Positive Integer Power: If n is a positive integer, then:

$\frac{d}{dx} x^n = n x^{n-1}$

• For all x where the powers are defined.

Example 1

• Differentiate the following powers of x:

• (a) $x^1$ (b) $x^n$ (c) $x^{-1}$ (d) $1/x$ (e) $x^{1/2}$ (f) $\sqrt{x}$

• Constant Multiple Rule: If u is a differentiable function and c is a constant, then:

$\frac{d}{dx} [c u] = c \frac{du}{dx}$

• Negative of a Function: The derivative of the negative of a function is the negative of the derivative.

$\frac{d}{dx} [-u] = -\frac{du}{dx}$

• Sum Rule: If u and v are differentiable functions, then:

$\frac{d}{dx} [u + v] = \frac{du}{dx} + \frac{dv}{dx}$

• Difference Rule: The derivative of a difference is the difference of the derivatives.

$\frac{d}{dx} [u - v] = \frac{du}{dx} - \frac{dv}{dx}$

Example 2

• (a) The derivative formula for $y = 3x^2$ is $\frac{d}{dx} 3x^2 = 3 \cdot 2x = 6x$

• (b) The derivative of the negative of a function: $\frac{d}{dx} [-u] = -\frac{du}{dx}$

Additional info:

• These notes cover the foundational concepts of derivatives, including their geometric meaning, formal definition, and basic rules for computation. The examples and figures illustrate how to apply these concepts to specific functions and interpret their graphical behavior.