Section 3.3: Differentiation Rules – Comprehensive Study Notes

Section 3.3 Differentiation Rules

Definition of the Derivative

The derivative of a function f(x) with respect to the variable x is a fundamental concept in calculus, representing the instantaneous rate of change of the function at a given point. The derivative is defined as:

• Definition: The derivative of f(x) at x is given by the limit:

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

• Existence: The limit must exist for the derivative to be defined at x.

One-Sided Derivatives at Endpoints

At the endpoints of a closed interval, derivatives are defined using one-sided limits. This is necessary because the function may not be defined on both sides of the endpoint.

• Left Endpoint: The derivative at a uses the right-hand limit.

• Right Endpoint: The derivative at b uses the left-hand limit.

$\text{Slope at } a: \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$ $\text{Slope at } b: \lim_{h \to 0^-} \frac{f(b+h) - f(b)}{h}$

Non-Differentiable Points

Some functions are not differentiable at certain points due to discontinuities, corners, cusps, or vertical tangent lines.

• Corner: The function y = |x| is not differentiable at x = 0 because the left and right derivatives are not equal.

• Cusp: The slope approaches infinity from opposite directions.

• Vertical Tangent: The slope approaches infinity from both sides.

Derivative of a Constant Function

A constant function has a derivative of zero everywhere, since its value does not change.

• Formula: If f(x) = c, then:

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

Graphical Interpretation

The slope of a horizontal line (constant function) is zero at every point.

Power Rule for Derivatives

The power rule is a fundamental rule for differentiating functions of the form xn, where n is any real number.

• Formula:

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

• Valid for all x where x^n and x^{n-1} are defined.

Derivative Constant Multiple Rule

If u is a differentiable function and c is a constant, the derivative of the product is the constant times the derivative of the function.

• Formula:

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

Derivative Sum Rule

The sum rule states that the derivative of the sum of two differentiable functions is the sum of their derivatives.

• Formula:

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

Derivative Product Rule

The product rule is used to differentiate the product of two functions.

• Formula:

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

Derivative Quotient Rule

The quotient rule is used to differentiate the quotient of two functions, provided the denominator is not zero.

• Formula:

$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$

Summary Table: Differentiation Rules

Rule	Formula
Constant Rule	$\frac{d}{dx} (c) = 0$
Power Rule	$\frac{d}{dx} (x^n) = n x^{n-1}$
Constant Multiple Rule	$\frac{d}{dx} (c u) = c \frac{du}{dx}$
Sum Rule	$\frac{d}{dx} (u + v) = \frac{du}{dx} + \frac{dv}{dx}$
Product Rule	$\frac{d}{dx} (u v) = u \frac{dv}{dx} + v \frac{du}{dx}$
Quotient Rule	$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$

Examples and Applications

• Example 1: Differentiate f(x) = x^3 using the power rule:

$\frac{d}{dx} (x^3) = 3x^2$

• Example 2: Differentiate f(x) = 5x^2 using the constant multiple rule:

$\frac{d}{dx} (5x^2) = 5 \cdot 2x = 10x$

• Example 3: Differentiate f(x) = x^2 + 3x using the sum rule:

$\frac{d}{dx} (x^2 + 3x) = 2x + 3$

• Example 4: Differentiate f(x) = x^2 \cdot x^3 using the product rule:

$\frac{d}{dx} (x^2 x^3) = x^2 \frac{d}{dx} (x^3) + x^3 \frac{d}{dx} (x^2) = x^2 \cdot 3x^2 + x^3 \cdot 2x = 3x^4 + 2x^4 = 5x^4$

• Example 5: Differentiate f(x) = \frac{x^2}{x^3} using the quotient rule:

$\frac{d}{dx} \left( \frac{x^2}{x^3} \right) = \frac{x^3 \cdot 2x - x^2 \cdot 3x^2}{(x^3)^2} = \frac{2x^4 - 3x^4}{x^6} = \frac{-x^4}{x^6} = -\frac{1}{x^2}$

Additional info: Academic context and examples were added to ensure completeness and clarity for exam preparation.