Chapter 3: Derivatives – Tangent Lines, Rates of Change, and Differentiation Rules

Section 3.1 Tangent Lines and the Derivative at a Point

Finding a Tangent Line to the Graph of a Function

The tangent line to a curve at a given point provides the best linear approximation to the curve near that point. The concept of the derivative is rooted in the idea of the slope of this tangent line.

• Slope of the Tangent Line: At a point $P(x_0, f(x_0))$ on the curve $y = f(x)$, the slope $m$ of the tangent line is defined as

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

• Tangent Line: The tangent line at $P$ is the line through $P$ with this slope.

Example

• Given $y = \frac{1}{x}$, find the slope at any point $x = a \neq 0$ and specifically at $x = -1$.

• Find where the slope equals $-\frac{1}{4}$.

• Analyze how the tangent line changes as $a$ varies.

Solution

• For $f(x) = \frac{1}{x}$, the slope at $(a, \frac{1}{a})$ is

$$ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim_{h \to 0} \frac{\frac{1}{a+h} - \frac{1}{a}}{h} = \lim_{h \to 0} \frac{-h}{h a(a+h)} = -\frac{1}{a^2}$$

• At $a = -1$, the slope is $-\frac{1}{(-1)^2} = -1$.

• The slope is $-\frac{1}{4}$ when $a^2 = 4$, so $a = 2$ or $a = -2$.

• As $a \to 0^+$, the slope approaches $-\infty$ and the tangent line becomes increasingly steep.

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 $f$ at $x_0$ is

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

• Example: For $y = 16t^2$, the speed of a falling rock at $t = 1$ is

$$ \lim_{h \to 0} \frac{16(1 + h)^2 - 16(1)^2}{h} = \lim_{h \to 0} \frac{16(h^2 + 2h + 1) - 16}{h} = \lim_{h \to 0} \frac{32h + 16h^2}{h} = 32$$

So, the speed at $t = 1$ is 32 ft/sec.

Summary of Interpretations

• The limit of the difference quotient represents:

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

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

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

  • The derivative $f'(x_0)$

Section 3.2 The Derivative as a Function

The Derivative as a Function

The derivative can be viewed as a function that assigns to each $x$ the value of the derivative at $x$.

• Definition:

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

• Alternative form:

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

Calculating Derivatives from the Definition

• Example: Differentiate $f(x) = \frac{x}{x-1}$.

• Solution:

$$ \begin{align*} f(x) &= \frac{x}{x-1} \\ f(x+h) &= \frac{x+h}{x+h-1} \\ f'(x) &= \lim_{h \to 0} \frac{\frac{x+h}{x+h-1} - \frac{x}{x-1}}{h} \\ &= \lim_{h \to 0} \frac{-h}{h(x-1)(x+h-1)} \\ &= -\frac{1}{(x-1)^2} \end{align*}$$

Notation

• Common notations for the derivative include:

$$ f'(x),\ y',\ \frac{dy}{dx},\ \frac{df}{dx},\ D_x f(x)$$

• To indicate the value at $x = a$:

$$ f'(a) = \left. \frac{dy}{dx} \right|_{x=a} = \left. \frac{df}{dx} \right|_{x=a} = \left. \frac{d}{dx} f(x) \right|_{x=a}$$

Graphing the Derivative

• To graph $y = f'(x)$, plot the slopes of tangent lines to $y = f(x)$ at various points and connect these points smoothly.

Differentiability and One-Sided Derivatives

Differentiable on an Interval; One-Sided Derivatives

• A function $y = f(x)$ is differentiable on an open interval if it has a derivative at every point in the interval.

• On a closed interval $[a, b]$, $f$ is differentiable if it is differentiable on $(a, b)$ and the following limits exist:

$$ \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h} \quad \text{(right-hand derivative at$a$)}$$\lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h} \quad \text{(left-hand derivative at$b$)}$$

• Example: The square root function $y = \sqrt{x}$ is not differentiable at $x = 0$ because the tangent is vertical there.

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

• A function fails to have a derivative at a point if:

  • There is a corner (one-sided derivatives differ)

  • There is a cusp (slopes approach $\infty$ from one side and $-\infty$ from the other)

  • There is a vertical tangent (slopes approach $\infty$ from both sides)

  • There is a discontinuity

  • There is wild oscillation

Differentiable Functions Are Continuous

• Theorem: If $f$ has a derivative at $x = c$, then $f$ is continuous at $x = c$.

• Caution: The converse is not true; a function can be continuous but not differentiable at a point.

Section 3.3 Differentiation Rules

Powers, Multiples, Sums, and Differences

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

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

• Derivative of a Power: For any real $n$:

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

• Examples:

$$ \frac{d}{dx} x^3 = 3x^2$$\frac{d}{dx} x^{2/3} = \frac{2}{3} x^{-1/3}$$\frac{d}{dx} x^{\sqrt{2}} = \sqrt{2} x^{\sqrt{2} - 1}$$\frac{d}{dx} \frac{1}{x^4} = -4x^{-5} = -\frac{4}{x^5}$$\frac{d}{dx} x^{-4/3} = -\frac{4}{3} x^{-7/3}$$\frac{d}{dx} \sqrt{x^{2+\pi}} = \frac{1}{2}(2+\pi)x^{(2+\pi)/2 - 1} = \frac{1}{2}(2+\pi)x^{(\pi/2)}$$

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

$$ \frac{d}{dx} (cu) = c \frac{du}{dx}$$

• Sum Rule: If $u$ and $v$ are differentiable,

$$ \frac{d}{dx} (u + v) = \frac{du}{dx} + \frac{dv}{dx}$$

• Example: For $y = 3x^2 + 4x - 5$,

$$ \frac{dy}{dx} = 6x + 4$$

Additional info: The slides continue with further differentiation rules, including the Product and Quotient Rules, and applications to exponential and trigonometric functions.