2.5 Basic Differentiation Rules Homework Answer Key Pdf

2.5 Basic Differentiation Rules: Homework Answer Key & Beyond

Finding a readily available PDF answer key for a specific textbook's homework problems can be tricky. Copyright restrictions and the varied editions of textbooks make a universal answer key impossible. However, this comprehensive guide will cover the 2.5 basic differentiation rules in detail, providing you with the knowledge and examples to solve any problem related to these rules, effectively acting as your own personalized answer key. We will go beyond simply providing answers, focusing on understanding the why behind the calculations, solidifying your understanding of differential calculus.

This article will cover the core differentiation rules, providing examples and explanations to enhance your understanding. Mastering these rules is crucial for progressing to more advanced calculus concepts.

1. The Constant Rule

The constant rule states that the derivative of a constant function is always zero. If f(x) = c, where 'c' is a constant, then f'(x) = 0.

Why? A constant function represents a horizontal line. The slope of a horizontal line is always zero, and the derivative represents the instantaneous slope of a function at any given point.

Example:

• Problem: Find the derivative of f(x) = 5.
• Solution: f'(x) = 0

2. The Power Rule

The power rule is fundamental and widely used. If f(x) = x^n, where 'n' is a constant, then f'(x) = nx^(n-1).

Why? The power rule arises from the definition of the derivative as a limit. The derivation involves algebraic manipulation and limit properties, which are covered in more advanced calculus courses. For now, focusing on applying the rule is sufficient.

Examples:

• Problem: Find the derivative of f(x) = x^3.

• Solution: f'(x) = 3x^(3-1) = 3x^2

• Problem: Find the derivative of f(x) = x. (Remember, x = x¹)

• Solution: f'(x) = 1x^(1-1) = 1x^0 = 1

• Problem: Find the derivative of f(x) = 1/x (Rewrite as x⁻¹).

• Solution: f'(x) = -1x^(-1-1) = -x⁻² = -1/x²

• Problem: Find the derivative of f(x) = √x (Rewrite as x¹/²)

• Solution: f'(x) = (1/2)x^((1/2)-1) = (1/2)x⁻¹/² = 1/(2√x)

3. The Constant Multiple Rule

If f(x) = cg(x), where 'c' is a constant, then f'(x) = cg'(x). This rule states that you can pull a constant factor out before differentiating.

Why? This rule is a consequence of the linearity property of derivatives. The derivative is a linear operator, meaning it distributes over addition and commutes with scalar multiplication.

Examples:

• Problem: Find the derivative of f(x) = 7x^4.

• Solution: f'(x) = 7 * (4x³) = 28x³

• Problem: Find the derivative of f(x) = -3/x² (Rewrite as -3x⁻²)

• Solution: f'(x) = -3 * (-2x⁻³) = 6x⁻³ = 6/x³

4. The Sum/Difference Rule

If f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x). The derivative of a sum or difference is the sum or difference of the derivatives.

Why? This rule is another direct consequence of the linearity of the derivative operator.

Examples:

• Problem: Find the derivative of f(x) = 3x² + 5x - 2.

• Solution: f'(x) = 6x + 5

• Problem: Find the derivative of f(x) = x³ - 4x² + 7x - 9

• Solution: f'(x) = 3x² - 8x + 7

• Problem: Find the derivative of f(x) = √x + 1/x (Rewrite as x¹/² + x⁻¹)

• Solution: f'(x) = (1/(2√x)) - (1/x²)

5. Combining the Rules: Advanced Examples

Many problems will require you to combine these basic rules. Let's tackle some more complex examples:

Example 1:

Problem: Find the derivative of f(x) = 2x³(x² + 4x - 1)

Solution: First, expand the expression:

f(x) = 2x⁵ + 8x⁴ - 2x³

Now differentiate term by term using the power rule and constant multiple rule:

f'(x) = 10x⁴ + 32x³ - 6x²

Example 2:

Problem: Find the derivative of f(x) = (x² + 3)/(x)

Solution: First, simplify the expression by dividing each term in the numerator by the denominator:

f(x) = x + 3x⁻¹

Now differentiate:

f'(x) = 1 - 3x⁻² = 1 - 3/x²

Example 3:

Problem: Find the derivative of f(x) = (x² + 2x + 1)(x - 1)

Solution: You can either expand the expression first or use the product rule (which will be introduced in a more advanced section). Let's expand:

f(x) = x³ + 2x² + x - x² - 2x -1 = x³ + x² - x - 1

Now differentiate:

f'(x) = 3x² + 2x - 1

Beyond the Basics: A Glimpse into Advanced Differentiation

While the 2.5 basic rules form a solid foundation, calculus introduces more advanced rules to tackle more complex functions. These include:

• The Product Rule: Used for differentiating the product of two functions. If f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).
• The Quotient Rule: Used for differentiating the quotient of two functions. If f(x) = g(x)/h(x), then f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]².
• The Chain Rule: Used for differentiating composite functions (functions within functions). If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
• Implicit Differentiation: Used for differentiating equations where y is not explicitly defined as a function of x.
• Logarithmic Differentiation: A technique used to simplify the differentiation of functions with exponents or products.

Mastering the 2.5 basic differentiation rules is the key to unlocking these more advanced techniques. Practice consistently, work through numerous examples, and don't hesitate to seek clarification when needed. Remember, understanding the underlying principles is more important than simply memorizing formulas. This deep understanding will be invaluable as you progress through your calculus studies. This detailed guide, while not a downloadable PDF, serves as a comprehensive resource for understanding and applying these crucial differentiation rules. Use it to build a strong foundation in calculus!