Calculus 1: Derivatives Rules Derivatives of eˣ and ln x ddxeˣeˣ ddxln x1x

What Is This?

Derivatives of eˣ and ln x are fundamental concepts in calculus. d/dx[eˣ]=eˣ means the derivative of the exponential function eˣ is itself, eˣ. d/dx[ln x]=1/x means the derivative of the natural logarithm function ln x is 1/x. These topics appear in exams to test your understanding of differentiation rules and your ability to apply them to various functions.

Why It Matters

This topic is tested in calculus exams, including AP Calculus, IB Mathematics, and university-level calculus courses. It frequently appears and typically carries moderate to high marks. It tests your ability to differentiate functions, understand exponential and logarithmic properties, and apply these concepts to solve problems.

Core Concepts

• Exponential Function: The function eˣ where e is Euler's number (approximately 2.71828).
• Natural Logarithm: The function ln x, which is the logarithm to the base e.
• Derivative: The rate of change of a function, denoted by d/dx.
• Chain Rule: A rule for differentiating compositions of functions.
• Product and Quotient Rules: Rules for differentiating products and quotients of functions.

Prerequisites

• Understanding of basic differentiation rules.
• Knowledge of exponential and logarithmic functions.
• Familiarity with the chain rule.

The Rule-Book (How It Works)

Primary Rule

• The derivative of eˣ is eˣ.
• The derivative of ln x is 1/x.

Sub-rules and Edge Cases

• For eˣ: No exceptions; it holds for all real x.
• For ln x: Defined only for x > 0.

Visual Pattern

• eˣ differentiates to itself: eˣ.
• ln x differentiates to its reciprocal: 1/x.

Exam / Job / Audit Weighting

• Frequency: High
• Difficulty Rating: Intermediate
• Question Type: Multiple choice, short answer, problem-solving

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. d/dx[eˣ] = eˣ
2. d/dx[ln x] = 1/x
3. Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x)

Worked Examples (Step-by-Step)

Easy

Question: Find the derivative of f(x) = eˣ.

Step-by-Step: 1. Recognize the function f(x) = eˣ.
2. Apply the rule d/dx[eˣ] = eˣ.

Answer: f'(x) = eˣ

Medium

Question: Find the derivative of f(x) = ln(3x).

Step-by-Step: 1. Recognize the function f(x) = ln(3x).
2. Apply the chain rule: d/dx[ln(u)] = 1/u * du/dx.
3. Let u = 3x, then du/dx = 3.
4. Substitute: f'(x) = 1/(3x) * 3 = 1/x.

Answer: f'(x) = 1/x

Hard

Question: Find the derivative of f(x) = e^(ln x).

Step-by-Step: 1. Recognize the function f(x) = e^(ln x).
2. Apply the chain rule: d/dx[e^(u)] = e^(u) * du/dx.
3. Let u = ln x, then du/dx = 1/x.
4. Substitute: f'(x) = e^(ln x) * 1/x = x * 1/x = 1.

Answer: f'(x) = 1

Common Exam Traps & Mistakes

1. Mistake: Forgetting the chain rule for ln(3x).
2. Wrong Answer: 1/3x.

3. Correct Approach: Use the chain rule to get 1/x.

4. Mistake: Applying the derivative of eˣ incorrectly to e^(2x).

5. Wrong Answer: e^(2x).

6. Correct Approach: Use the chain rule to get 2e^(2x).

7. Mistake: Not recognizing the domain of ln x.

8. Wrong Answer: 1/x for x ≤ 0.
9. Correct Approach: ln x is only defined for x > 0.

Shortcut Strategies & Exam Hacks

• Memory Aid: "eˣ stays the same, ln x goes to 1/x."
• Pattern Recognition: Look for eˣ and ln x in composite functions to apply the chain rule quickly.
• Elimination Strategy: If a choice doesn't match the derivative rules, eliminate it.

Question-Type Taxonomy

1. Multiple Choice: Direct application of derivative rules.
2. Example: What is the derivative of e^(2x)?

3. Favored By: AP Calculus, IB Mathematics.

4. Short Answer: Calculate the derivative of a given function.

5. Example: Find f'(x) if f(x) = ln(5x).

6. Favored By: University-level calculus exams.

7. Problem-Solving: Apply derivatives to solve real-world problems.

8. Example: Use the derivative of eˣ to model population growth.
9. Favored By: Applied mathematics courses.

Practice Set (MCQs)

Question 1

Question: What is the derivative of f(x) = e^(3x)?

Options: A. 3e^(3x)
B. e^(3x)
C. 3x * e^(3x)
D. e^(x)

Correct Answer: A. 3e^(3x)

Explanation: Apply the chain rule: d/dx[e^(3x)] = e^(3x) * 3.

Why the Distractors Are Tempting: - B: Forgets the chain rule.
- C: Incorrect application of the chain rule.
- D: Misapplies the derivative of eˣ.

Question 2

Question: What is the derivative of f(x) = ln(2x)?

Options: A. 1/2x
B. 2/x
C. 1/x
D. 2x

Correct Answer: C. 1/x

Explanation: Apply the chain rule: d/dx[ln(2x)] = 1/(2x) * 2 = 1/x.

Why the Distractors Are Tempting: - A: Forgets the chain rule.
- B: Incorrect application of the chain rule.
- D: Misapplies the derivative of ln x.

Question 3

Question: What is the derivative of f(x) = e^(ln(2x))?

Options: A. 2x
B. 2
C. 1/x
D. 2e^(ln(2x))

Correct Answer: B. 2

Explanation: Apply the chain rule: d/dx[e^(ln(2x))] = e^(ln(2x)) * 1/(2x) * 2 = 2.

Why the Distractors Are Tempting: - A: Misapplies the chain rule.
- C: Incorrect application of the chain rule.
- D: Overcomplicates the derivative.

Question 4

Question: What is the derivative of f(x) = ln(eˣ)?

Options: A. x
B. 1/eˣ
C. eˣ
D. 1

Correct Answer: D. 1

Explanation: Apply the chain rule: d/dx[ln(eˣ)] = 1/(eˣ) * eˣ = 1.

Why the Distractors Are Tempting: - A: Misapplies the chain rule.
- B: Incorrect application of the chain rule.
- C: Overcomplicates the derivative.

Question 5

Question: What is the derivative of f(x) = e^(x^2)?

Options: A. 2x * e^(x^2)
B. e^(x^2)
C. 2e^(x^2)
D. x * e^(x^2)

Correct Answer: A. 2x * e^(x^2)

Explanation: Apply the chain rule: d/dx[e^(x^2)] = e^(x^2) * 2x.

Why the Distractors Are Tempting: - B: Forgets the chain rule.
- C: Incorrect application of the chain rule.
- D: Misapplies the chain rule.

30-Second Cheat Sheet

• d/dx[eˣ] = eˣ
• d/dx[ln x] = 1/x
• Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
• eˣ differentiates to itself
• ln x differentiates to its reciprocal
• ln x is defined only for x > 0

Learning Path

1. Beginner Foundation: Review basic differentiation rules and properties of exponential and logarithmic functions.
2. Core Rules: Memorize d/dx[eˣ] = eˣ and d/dx[ln x] = 1/x.
3. Practice: Solve simple derivative problems.
4. Timed Drills: Practice under exam conditions.
5. Mock Tests: Take full-length practice exams.

Related Topics

1. Chain Rule: Often used alongside derivatives of eˣ and ln x.
2. Product and Quotient Rules: Applied in more complex derivative problems.
3. Implicit Differentiation: Involves differentiating functions with eˣ and ln x terms.