04-05 - Derivatives of Elementary Functions
Phase: 4 | Subject: 04-05 Prerequisites: 04-04-differentiation-rules.md Next subject: 04-06-implicit-differentiation.md
Learning Objectives
By the end of this subject, you will be able to:
- Differentiate polynomials, trigonometric, exponential, and logarithmic functions
- Apply the chain rule with these elementary functions
- Recognise standard derivative patterns
Core Content
Polynomials
$d/dx[xⁿ] = n·x^(n-1)$
Example: d/dx[3x⁴ - 2x² + 5] = 12x³ - 4x
Trigonometric Functions
$$ d/dx[sin(x)] = cos(x) d/dx[cos(x)] = -sin(x) d/dx[tan(x)] = sec²(x) d/dx[csc(x)] = -csc(x)·cot(x) d/dx[sec(x)] = sec(x)·tan(x) d/dx[cot(x)] = -csc²(x) $$
Memory tips: - sin and cos swap with a sign change for cos - tan and cot have squared results with opposite signs - sec and csc multiply by themselves and tan/cot respectively
Example: d/dx[sin(2x)] = 2·cos(2x) (chain rule) Example: d/dx[cos(x³)] = -3x²·sin(x³)
Exponential Functions
$$ d/dx[eˣ] = eˣ d/dx[aˣ] = aˣ·ln(a) $$
⚠️ THIS IS CRITICAL — eˣ is the only (non-zero) function that is its own derivative. This property makes e the natural base for calculus and appears everywhere: differential equations, probability (normal distribution), complex numbers, and neural network activation functions.
Example: d/dx[e^(3x)] = 3e^(3x) Example: d/dx[2ˣ] = 2ˣ·ln(2)
Logarithmic Functions
$$ d/dx[ln(x)] = 1/x d/dx[logₐ(x)] = 1/(x·ln(a)) $$
Example: d/dx[ln(x² + 1)] = 2x/(x² + 1) (chain rule)
Inverse Trigonometric Functions
$$ d/dx[arcsin(x)] = 1/√(1 - x²) d/dx[arccos(x)] = -1/√(1 - x²) d/dx[arctan(x)] = 1/(1 + x²) d/dx[arccot(x)] = -1/(1 + x²) d/dx[arcsec(x)] = 1/(|x|√(x² - 1)) d/dx[arccsc(x)] = -1/(|x|√(x² - 1)) $$
Memory tip: arcsin and arccos are negatives of each other; arctan and arccot the same. Notice the pattern: arcsin/arccos involve √(1-x²), arctan/arccot involve (1+x²), arcsec/arccsc involve |x|√(x²-1).
Example: d/dx[arcsin(3x)] = 3/√(1 - 9x²) (chain rule) Example: d/dx[arctan(x²)] = 2x/(1 + x⁴)
Worked Examples
Example: d/dx[x³·sin(x)] Product rule: 3x²·sin(x) + x³·cos(x)
Example: d/dx[eˣ·cos(x)] Product rule: eˣ·cos(x) + eˣ·(-sin(x)) = eˣ(cos(x) - sin(x))
Example: d/dx[ln(sin(x))] Chain: (1/sin(x))·cos(x) = cot(x)
Key Terms
- 04 05 Derivatives Of Elementary Functions
- Combined Examples
- Correct: B)
- Correct: C)
- Exponential Functions
- Inverse Trigonometric Functions
- Logarithmic Functions
- Polynomials
- Trigonometric Functions
Quiz
Q1: What does the concept of Exponential Functions primarily refer to in this subject?
A) The definition and application of Exponential Functions B) A computational error related to Exponential Functions C) A visual representation of Exponential Functions D) A historical anecdote about Exponential Functions
Correct: A)
- If you chose A: Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus. Correct!
- If you chose B: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.
- If you chose C: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.
- If you chose D: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.
Q2: Which of the following is the key formula discussed in this subject?
A) An unrelated formula from a different topic B) d/dx[xⁿ] = n·x^(n-1) C) The inverse operation of the formula in question D) A simplified version of d/dx[xⁿ] = n·x^(n-1)...
Correct: B)
- If you chose A: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
- If you chose B: The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated. Correct!
- If you chose C: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
- If you chose D: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
Q3: What is the primary purpose of Inverse Trigonometric Functions?
A) It is used to inverse trigonometric functions in mathematical analysis B) It is primarily a historical notation system C) It replaces all other methods in this domain D) It is used only in advanced research contexts
Correct: A)
- If you chose A: Inverse Trigonometric Functions serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
- If you chose B: This is incorrect. Inverse Trigonometric Functions serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose C: This is incorrect. Inverse Trigonometric Functions serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose D: This is incorrect. Inverse Trigonometric Functions serves the purpose described in the correct answer. The other options misrepresent its role.
Q4: Which statement about Logarithmic Functions is TRUE?
A) Logarithmic Functions is an advanced topic beyond this subject's scope B) Logarithmic Functions is mentioned only as a historical footnote C) Logarithmic Functions is not related to this subject D) Logarithmic Functions is a fundamental concept covered in this subject
Correct: D)
- If you chose A: This is incorrect. Logarithmic Functions is a fundamental concept covered in this subject. This subject covers Logarithmic Functions as part of its core content.
- If you chose B: This is incorrect. Logarithmic Functions is a fundamental concept covered in this subject. This subject covers Logarithmic Functions as part of its core content.
- If you chose C: This is incorrect. Logarithmic Functions is a fundamental concept covered in this subject. This subject covers Logarithmic Functions as part of its core content.
- If you chose D: Logarithmic Functions is a fundamental concept covered in this subject. This subject covers Logarithmic Functions as part of its core content. Correct!
Q5: Based on the worked examples in this subject, what is the correct result?
A) The inverse of the correct answer B) An unrelated numerical value C) A different result from a common mistake D) 2·cos(2x) (chain rule)
Correct: D)
- If you chose A: This is incorrect. The worked examples show that the result is 2·cos(2x) (chain rule). The other options represent common errors.
- If you chose B: This is incorrect. The worked examples show that the result is 2·cos(2x) (chain rule). The other options represent common errors.
- If you chose C: This is incorrect. The worked examples show that the result is 2·cos(2x) (chain rule). The other options represent common errors.
- If you chose D: The worked examples show that the result is 2·cos(2x) (chain rule). The other options represent common errors. Correct!
Q6: How are Logarithmic Functions and Polynomials related?
A) Logarithmic Functions and Polynomials are closely related concepts B) Logarithmic Functions is the inverse of Polynomials C) Logarithmic Functions and Polynomials are completely unrelated topics D) Logarithmic Functions is a special case of Polynomials
Correct: A)
- If you chose A: Both Logarithmic Functions and Polynomials are covered in this subject as interconnected topics. Correct!
- If you chose B: This is incorrect. Both Logarithmic Functions and Polynomials are covered in this subject as interconnected topics.
- If you chose C: This is incorrect. Both Logarithmic Functions and Polynomials are covered in this subject as interconnected topics.
- If you chose D: This is incorrect. Both Logarithmic Functions and Polynomials are covered in this subject as interconnected topics.
Q7: What is a common pitfall when working with Trigonometric Functions?
A) A common mistake is confusing Trigonometric Functions with a similar concept B) The main error with Trigonometric Functions is using it when it is not needed C) Trigonometric Functions is always computed the same way in all contexts D) Trigonometric Functions has no common misconceptions
Correct: A)
- If you chose A: Students often confuse Trigonometric Functions with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
- If you chose B: This is incorrect. Students often confuse Trigonometric Functions with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose C: This is incorrect. Students often confuse Trigonometric Functions with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose D: This is incorrect. Students often confuse Trigonometric Functions with similar-sounding or related concepts. Pay attention to the precise definitions.
Q8: When should you apply Common Pitfalls?
A) Common Pitfalls is not practically useful B) Use Common Pitfalls only in pure mathematics contexts C) Apply Common Pitfalls to solve problems in this subject's domain D) Avoid Common Pitfalls unless explicitly instructed
Correct: C)
- If you chose A: This is incorrect. Common Pitfalls is a practical tool used throughout this subject to solve relevant problems.
- If you chose B: This is incorrect. Common Pitfalls is a practical tool used throughout this subject to solve relevant problems.
- If you chose C: Common Pitfalls is a practical tool used throughout this subject to solve relevant problems. Correct!
- If you chose D: This is incorrect. Common Pitfalls is a practical tool used throughout this subject to solve relevant problems.
Practice Problems
-
d/dx[cos(3x)] Answer: -3·sin(3x)
-
d/dx[e^(x²)] Answer: 2x·e^(x²)
-
d/dx[ln(2x + 1)] Answer: 2/(2x + 1)
-
d/dx[x·tan(x)] Answer: tan(x) + x·sec²(x)
-
d/dx[sin²(x)] Answer: 2·sin(x)·cos(x) = sin(2x)
-
d/dx[arctan(2x)] Answer: Chain rule: 2/(1 + (2x)²) = 2/(1 + 4x²)
Summary
Key takeaways:
- d/dx[xⁿ] = n·x^(n-1)
- sin' = cos, cos' = -sin, tan' = sec²
- d/dx[eˣ] = eˣ (its own derivative!)
- d/dx[aˣ] = aˣ·ln(a)
- d/dx[ln(x)] = 1/x
- d/dx[arcsin(x)] = 1/√(1 - x²), d/dx[arctan(x)] = 1/(1 + x²)
- Always apply chain rule when the argument is not just x
Pitfalls
- Forgetting the chain rule. d/dx[sin(3x)] = cos(3x)·3, NOT just cos(3x). Every time the argument is more than x, the chain rule multiplies by the inner derivative.
- Mixing up derivative signs. sin' = cos, but cos' = -sin (the negative sign is easy to drop). Similarly, cot' = -csc², not +csc².
- Confusing exponential rules. d/dx[aˣ] = aˣ·ln(a), NOT x·a^(x-1). The power rule d/dx[xⁿ] = n·x^(n-1) only applies when the base is the variable, not the exponent.
- Forgetting the absolute value in arcsec/arccsc derivatives. d/dx[arcsec(x)] = 1/(|x|√(x²-1)), not 1/(x√(x²-1)). The absolute value matters for negative x.
- Missing simplification opportunities. For ln(x³), simplify to 3·ln(x) first (derivative 3/x), rather than using the chain rule and needing to simplify (1/x³)·3x² = 3/x anyway.
Next Steps
Next up: 04-06-implicit-differentiation.md