🔹 1. Derivative Definition (Conceptual Core)
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$
When it appears:
- FRQs asking for derivative from first principles
- Justifying differentiability
Tip: Be comfortable expanding expressions algebraically.
🔹 2. Power Rule (Most Used Rule)
$$\frac{d}{dx} x^n = n x^{n-1}$$
When it appears:
- Everywhere (MCQ + FRQ)
- Polynomial differentiation
🔹 3. Product Rule
$$(fg)’ = f’g + fg’$$
When it appears:
- Functions multiplied together
- Common in velocity/acceleration problems
🔹 4. Chain Rule (Critical for BC)
$$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$
When it appears:
- Composite functions
- Exponentials, trig, implicit differentiation
👉 One of the most tested rules in both AB and BC
🔹 5. Implicit Differentiation
Used when variables are mixed:
Example:
$$x^2 + y^2 = 25$$
When it appears:
- FRQs involving curves not solved for (y)
🔹 6. Derivative of Exponential Functions
$$\frac{d}{dx} e^x = e^x$$
$$\frac{d}{dx} a^x = a^x \ln a$$
When it appears:
- Growth/decay models
- BC-heavy questions
🔹 7. Derivative of Logarithmic Functions
$$\frac{d}{dx} \ln x = \frac{1}{x}$$
When it appears:
- Log differentiation
- Simplifying complex expressions
🔹 8. Definite Integral (Fundamental Theorem of Calculus)
$$\int_a^b f(x),dx = F(b) – F(a)$$
When it appears:
- Area under a curve
- Accumulation functions
👉 This connects derivatives and integrals
🔹 9. Derivative of an Integral (FTC Part 1)
$$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$
When it appears:
- FRQs with accumulation functions
- Conceptual understanding of rates
🔹 10. Average Value of a Function
$$f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$$
When it appears:
- Guaranteed appearance in many exams
🔹 11. Integration by Parts (BC Only)
$$\int u \, dv = uv – \int v \, du$$
When it appears:
- Product of functions
- BC free-response questions
🔹 12. Logistic Differential Equation (BC High-Frequency)
$$\frac{dP}{dt} = kP\left(1 – \frac{P}{L}\right)$$
When it appears:
- Population growth modeling
- Interpretation-based FRQs
🔹 13. Slope of Tangent Line
$$m = f'(a)$$
When it appears:
- Tangent line equations
- Linear approximation problems
🔹 14. Linear Approximation
$$L(x) = f(a) + f'(a)(x – a)$$
When it appears:
- Estimation problems
- FRQs involving approximation
🎯 Final Strategy for the Exam
- Don’t just memorize formulas → know when to apply them
- Focus on:
- Chain Rule
- FTC
- Accumulation functions
- Practice FRQ-style explanations