DIFFERENTIATION TECHNIQUES
| Concept | IB Definition & Role |
|---|---|
| Trigonometric derivatives | Exact derivative rules for sin, cos, and tan used to model periodic motion such as waves. |
| Exponential & logarithmic derivatives | Rules governing growth, decay, and scaling models in science and economics. |
| Power rule (rational exponents) | Differentiation of roots, reciprocals, and inverse powers. |
| Chain, product, quotient rules | Structured methods for composite, multiplied, and divided functions. |
| Related rates | Applications of derivatives to time-dependent systems. |
📌 Derivatives of trigonometric functions
- Derivative of sin x: The derivative of sin x is cos x; for example, at x = 0, the slope of sin x equals cos 0 = 1.
- Derivative of cos x: The derivative of cos x is −sin x; for example, at x = π/2, the slope is −1.
- Derivative of tan x: The derivative of tan x is sec²x; for example, tan x increases rapidly near x = π/2.
- These derivatives must be memorised exactly, e.g. sin(3x) differentiates to 3cos(3x).
- They frequently appear combined with the chain rule in exam questions.
🌍 Real-World Connection
- Sinusoidal derivatives are used to compute velocity from displacement in wave motion.
- Electrical engineers use cos and sin derivatives in alternating current analysis.
- Acceleration in circular motion is found using trigonometric rates of change.
📌 Derivatives of exponential and logarithmic functions
- Derivative of eˣ: The derivative of eˣ is eˣ; for example, growth rate equals value at every point.
- Derivative of ln x: The derivative of ln x is 1/x; for example, at x = 2 the slope is 0.5.
- These functions appear in compound interest and population growth models.
- They often require the chain rule, e.g. ln(3x²).
- Domain restrictions must always be stated explicitly.
🧠 Examiner Tip
- Always state domain restrictions when differentiating ln x.
- Do not cancel terms before applying differentiation rules.
- Method marks are awarded for correct rule identification.
📌 Power rule for rational exponents
- The derivative of xⁿ is n·xⁿ⁻¹; for example, x¹ᐟ² becomes ½x⁻¹ᐟ².
- Negative powers must be handled carefully, e.g. x⁻² becomes −2x⁻³.
- All radicals should be rewritten as powers before differentiating.
- This rule applies to fractional, integer, and negative exponents.
- Errors often occur when radicals are left in root form.
🔍 TOK Perspective
- Why does a single algebraic rule apply universally to all rational exponents?
- Does abstraction increase certainty in mathematics?
- How do symbolic rules shape mathematical knowledge?
📌 Chain, product, and quotient rules
- The chain rule differentiates composite functions such as sin(x²).
- The product rule applies when functions are multiplied, e.g. x²eˣ.
- The quotient rule is used for ratios such as x² / ln x.
- Correct rule identification is essential for full marks.
- These rules are heavily tested at HL level.
📌 Related rates
- Related rates problems involve differentiation with respect to time.
- Variables are connected through geometric or physical equations.
- Known values are substituted at a specific instant.
- Units must be consistent throughout the solution.
- These questions test modelling and interpretation.
📐 IA Spotlight
- Related rates are ideal for modelling real-world motion.
- Clearly justify assumptions and variable relationships.
- Discuss limitations of continuous models.
📌 Exam-style questions
Multiple-choice questions
MCQ 1. What is the derivative of sin(3x)?
A. cos x B. 3cos(3x) C. cos(3x) D. −sin(3x)
Answer: B — Chain rule applied.
MCQ 2. What is the derivative of ln(x²)?
A. 1/x B. 2/x C. x² D. 2x
Answer: B — Differentiate using chain rule.
MCQ 3. What is the derivative of x⁻³?
A. −3x⁻⁴ B. 3x⁻² C. −x⁻⁴ D. x⁻²
Answer: A
MCQ 4. Which rule applies to y = x·eˣ?
A. Chain B. Quotient C. Product D. Power
Answer: C
Short-answer questions
SAQ 1. Why must radicals be rewritten as powers?
The power rule applies only to index notation, not radical form.
SAQ 2. Why is eˣ unique?
It is the only function equal to its own derivative.
Long-answer questions
LAQ 1.
(a) Differentiate y = sin(x²).
(b) State the rule used.
(c) Explain the structure of the derivative.
(a) dy/dx = 2x·cos(x²)
(b) Chain rule
(c) The outer derivative is multiplied by the derivative of the inner function.
LAQ 2.
(a) Differentiate y = (x²eˣ)/(ln x).
(b) Identify all rules used.
(c) State the domain of the function.
(a) Product and quotient rules are required.
(b) Product rule in numerator, quotient rule overall.
(c) x > 0 due to ln x.