Unit 3 - Notes

MTH005

Unit 3: Differentiation

1. Introduction to Derivatives

Differentiation is a fundamental tool in calculus concerned with calculating the instantaneous rate of change of quantities. Geometrically, the derivative represents the slope of the tangent line to a curve at a specific point.

1.1 Definition of the Derivative

The derivative of a function $f(x)$ with respect to $x$ is defined by the limit:

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

This formula is known as differentiation from first principles.

1.2 Notation

If $y = f(x)$ , the derivative can be denoted as:

$f'(x)$ (Lagrange notation)
$\frac{dy}{dx}$ (Leibniz notation)
$\frac{d}{dx}[f(x)]$ (Operator notation)

1.3 The Power Rule

For any real number $n$ , if $f(x) = x^n$ , then:

\frac{d}{dx}(x^n) = nx^{n-1}

Special Cases:

Derivative of a Constant: If $f(x) = c$ , then $f'(x) = 0$ .
Derivative of $x$ : If $f(x) = x$ , then $f'(x) = 1$ .
Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ .

2. Rules of Differentiation

When differentiating complex expressions, we utilize algebraic rules to break the function down into simpler parts.

2.1 Derivative of a Sum and Difference

The derivative of a sum or difference of functions is the sum or difference of their individual derivatives.

Formula:

\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)

Example:
Differentiate $y = 2x^3 + 5x - 7$ .

Step 1: Apply the rule term by term.

\frac{dy}{dx} = \frac{d}{dx}(2x^3) + \frac{d}{dx}(5x) - \frac{d}{dx}(7)

Step 2: Apply the Power Rule and Constant Rule.

\frac{dy}{dx} = 2(3x^{3-1}) + 5(1x^{1-1}) - 0

\frac{dy}{dx} = 6x^2 + 5

2.2 Derivative of a Product (Product Rule)

When a function is the product of two differentiable functions, $u(x)$ and $v(x)$ , we cannot simply multiply their derivatives. We must use the Product Rule.

Formula:

\frac{d}{dx} [u(x) \cdot v(x)] = u(x)v'(x) + v(x)u'(x)

Mnemonic: First times derivative of the second + Second times derivative of the first.

Example:
Differentiate $f(x) = (x^2 + 1)(3x - 2)$ .

Step 1: Identify $u$ and $v$ .
$u = x^2 + 1 \quad \Rightarrow \quad u' = 2x$
$v = 3x - 2 \quad \Rightarrow \quad v' = 3$

Step 2: Apply the formula.

f'(x) = (x^2 + 1)(3) + (3x - 2)(2x)

Step 3: Expand and Simplify.

f'(x) = 3x^2 + 3 + 6x^2 - 4x

f'(x) = 9x^2 - 4x + 3

2.3 Derivative of a Quotient (Quotient Rule)

When a function is the quotient of two differentiable functions, $\frac{u(x)}{v(x)}$ , we use the Quotient Rule.

Formula:

\frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}

Mnemonic: (Low $\cdot$ d-High) minus (High $\cdot$ d-Low) all over (Low squared).
Note: Order matters in the numerator because of the subtraction.

Example:
Differentiate $y = \frac{2x}{x^2 + 1}$ .

Step 1: Identify $u$ (top) and $v$ (bottom).
$u = 2x \quad \Rightarrow \quad u' = 2$
$v = x^2 + 1 \quad \Rightarrow \quad v' = 2x$

Step 2: Apply the formula.

\frac{dy}{dx} = \frac{(x^2 + 1)(2) - (2x)(2x)}{(x^2 + 1)^2}

Step 3: Simplify the numerator.

\frac{dy}{dx} = \frac{2x^2 + 2 - 4x^2}{(x^2 + 1)^2}

\frac{dy}{dx} = \frac{2 - 2x^2}{(x^2 + 1)^2}

3. Applications: Increasing and Decreasing Functions

Derivatives provide information about the behavior of a function's graph. Since the derivative represents the slope of the tangent, the sign of the derivative tells us if the function is rising or falling.

3.1 Definitions

Increasing Function: A function $f(x)$ is increasing on an interval if, for any two numbers $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) < f(x_2)$ . The graph goes up from left to right.
Decreasing Function: A function $f(x)$ is decreasing on an interval if, for any two numbers $x_1$ and $x_2$ in the interval, $x_1 < x_2$ implies $f(x_1) > f(x_2)$ . The graph goes down from left to right.

3.2 The First Derivative Test for Monotonicity

To determine the intervals where a function is increasing or decreasing without graphing it, we analyze the sign of $f'(x)$ .

Sign of Derivative	Behavior of $f(x)$	Geometric Interpretation
$f'(x) > 0$	Increasing	Tangent slope is positive
$f'(x) < 0$	Decreasing	Tangent slope is negative
$f'(x) = 0$	Stationary	Tangent is horizontal (turning point)

3.3 Steps to Find Intervals of Increase/Decrease

Find the derivative $f'(x)$ .
Find the Critical Points by setting $f'(x) = 0$ (or where undefined).
Plot these critical points on a number line to divide the domain into intervals.
Choose a test value inside each interval and plug it into $f'(x)$ .
Determine the sign (+ or -) to conclude if $f$ is increasing or decreasing on that interval.

3.4 Worked Example

Determine the intervals on which $f(x) = x^3 - 3x^2 - 9x + 5$ is increasing or decreasing.

Step 1: Find the derivative.

f'(x) = 3x^2 - 6x - 9

Step 2: Set $f'(x) = 0$ to find critical points.

3x^2 - 6x - 9 = 0

Divide by 3:

x^2 - 2x - 3 = 0

Factor:

(x - 3)(x + 1) = 0

Critical points:

x = 3

and

x = -1

Step 3: Create intervals.
The critical points divide the number line into three intervals:

$(-\infty, -1)$
$(-1, 3)$
$(3, \infty)$

Step 4: Test the intervals.

Interval $(-\infty, -1)$ : Test $x = -2$ .

$f'(-2) = 3(-2)^2 - 6(-2) - 9 = 12 + 12 - 9 = 15$
Sign: Positive (+) $\rightarrow$ Increasing.
Interval $(-1, 3)$ : Test $x = 0$ .

$f'(0) = 3(0)^2 - 6(0) - 9 = -9$
Sign: Negative (-) $\rightarrow$ Decreasing.
Interval $(3, \infty)$ : Test $x = 4$ .

$f'(4) = 3(4)^2 - 6(4) - 9 = 48 - 24 - 9 = 15$
Sign: Positive (+) $\rightarrow$ Increasing.

Conclusion:

$f(x)$ is increasing on $(-\infty, -1) \cup (3, \infty)$ .
$f(x)$ is decreasing on $(-1, 3)$ .

4. Summary Cheat Sheet

Rule Name	Formula
Power Rule	$\frac{d}{dx}(x^n) = nx^{n-1}$
Sum/Diff	$(u \pm v)' = u' \pm v'$
Product Rule	$(uv)' = uv' + vu'$
Quotient Rule	$\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2}$
Increasing	$f'(x) > 0$
Decreasing	$f'(x) < 0$

Unit 3 - Notes

Table of Contents

Unit 3: Differentiation

1. Introduction to Derivatives

1.1 Definition of the Derivative

1.2 Notation

1.3 The Power Rule

2. Rules of Differentiation

2.1 Derivative of a Sum and Difference

2.2 Derivative of a Product (Product Rule)

2.3 Derivative of a Quotient (Quotient Rule)

3. Applications: Increasing and Decreasing Functions

3.1 Definitions

3.2 The First Derivative Test for Monotonicity

3.3 Steps to Find Intervals of Increase/Decrease

3.4 Worked Example

4. Summary Cheat Sheet