AP Calculus BC derivatives free response questions

📚 Quick Study Guide

• 📈 Basic Differentiation Rules: Know your power rule, constant multiple rule, sum/difference rule, product rule, quotient rule, and chain rule. These are the foundation!
• ⛓️ Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Super important for composite functions!
• 📦 Product Rule: If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$. Don't forget it!
• ➗ Quotient Rule: If $y = \frac{u(x)}{v(x)}$, then $\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$. Remember the order!
• 🎯 Implicit Differentiation: Use this when you can't easily solve for $y$. Remember to apply the chain rule to $y$!
• 📐 Related Rates: Identify rates of change and use implicit differentiation to find relationships between them. Draw a picture!
• ✍️ Applications of Derivatives: Maxima, minima, concavity, points of inflection, curve sketching, and optimization problems. Understand what the derivative tells you about the function's behavior.

Practice Quiz

1. Question 1: What is the derivative of $f(x) = x^3 \sin(2x)$?
1. A) $3x^2 \cos(2x)$
2. B) $3x^2 \sin(2x) + 2x^3 \cos(2x)$
3. C) $3x^2 \sin(2x) - 2x^3 \cos(2x)$
4. D) $x^3 \cos(2x) + 3x^2 \sin(2x)$


2. Question 2: Find $\frac{dy}{dx}$ if $x^2 + y^2 = 25$.
1. A) $\frac{x}{y}$
2. B) $-\frac{x}{y}$
3. C) $\frac{y}{x}$
4. D) $-\frac{y}{x}$


3. Question 3: If $f(x) = \frac{x^2}{x+1}$, find $f'(1)$.
1. A) $\frac{1}{4}$
2. B) $\frac{3}{4}$
3. C) $1$
4. D) $\frac{1}{2}$


4. Question 4: What is the derivative of $f(x) = \ln(\cos(x))$?
1. A) $\tan(x)$
2. B) $-\tan(x)$
3. C) $\cot(x)$
4. D) $-\cot(x)$


5. Question 5: A spherical balloon is being inflated at a rate of $100 \pi \text{ cm}^3/\text{sec}$. Find the rate at which the radius is increasing when the radius is 5 cm. (Volume of a sphere: $V = \frac{4}{3}\pi r^3$)
1. A) 1 cm/sec
2. B) 2 cm/sec
3. C) 3 cm/sec
4. D) 4 cm/sec


6. Question 6: Find the second derivative of $f(x) = e^{x^2}$.
1. A) $2e^{x^2}$
2. B) $4x^2 e^{x^2}$
3. C) $(4x^2 + 2)e^{x^2}$
4. D) $(2x^2 + 4)e^{x^2}$


7. Question 7: What is the derivative of $f(x) = \arctan(3x)$?
1. A) $\frac{1}{1+9x^2}$
2. B) $\frac{3}{1+9x^2}$
3. C) $\frac{1}{1+3x^2}$
4. D) $\frac{3}{1+3x^2}$