High School Calculus Quotient Rule Exam Questions and Self-Assessment.

📚 Quick Study Guide

• ➗ The Quotient Rule is used to find the derivative of a function that is the ratio of two other functions.
• 📝 Formula: If $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$.
• ⚠️ Remember: The order of the terms in the numerator matters! It's (derivative of the top) times (bottom) minus (top) times (derivative of the bottom).
• 💡 Tip: Practice identifying $g(x)$ and $h(x)$ correctly before applying the rule.
• 🧠 Notation: $g'(x)$ and $h'(x)$ represent the derivatives of $g(x)$ and $h(x)$, respectively.
• 📈 Application: The quotient rule is essential for differentiating many complex functions in calculus.

🧪 Practice Quiz

1. What is the derivative of $f(x) = \frac{x^2}{x+1}$?
   1. $A) \frac{x^2 + 2x}{(x+1)^2}$
   2. $B) \frac{x^2 - 2x}{(x+1)^2}$
   3. $C) \frac{2x}{1}$
   4. $D) \frac{x^2 + 1}{(x+1)^2}$
2. Find the derivative of $f(x) = \frac{\sin(x)}{x}$.
   1. $A) \frac{x\cos(x) - \sin(x)}{x^2}$
   2. $B) \frac{\cos(x)}{1}$
   3. $C) \frac{x\cos(x) + \sin(x)}{x^2}$
   4. $D) \frac{\cos(x) - \sin(x)}{x^2}$
3. Calculate the derivative of $f(x) = \frac{e^x}{x^2}$.
   1. $A) \frac{e^x(x-2)}{x^3}$
   2. $B) \frac{e^x(x+2)}{x^4}$
   3. $C) \frac{e^x}{2x}$
   4. $D) \frac{e^x(x^2 - 2x)}{x^4}$
4. Determine $f'(x)$ for $f(x) = \frac{x^3}{\cos(x)}$.
   1. $A) \frac{3x^2\cos(x) + x^3\sin(x)}{\cos^2(x)}$
   2. $B) \frac{3x^2\cos(x) - x^3\sin(x)}{\cos^2(x)}$
   3. $C) \frac{3x^2}{-\sin(x)}$
   4. $D) \frac{3x^2}{\cos^2(x)}$
5. What is the derivative of $f(x) = \frac{\ln(x)}{x}$?
   1. $A) \frac{1 - \ln(x)}{x^2}$
   2. $B) \frac{\ln(x) - 1}{x^2}$
   3. $C) \frac{1}{x^2}$
   4. $D) \frac{1}{x}$
6. Find the derivative of $f(x) = \frac{x+2}{x-2}$.
   1. $A) \frac{-4}{(x-2)^2}$
   2. $B) \frac{4}{(x-2)^2}$
   3. $C) \frac{1}{1}$
   4. $D) \frac{-4}{(x+2)^2}$
7. Calculate $f'(x)$ for $f(x) = \frac{\tan(x)}{x}$.
   1. $A) \frac{x\sec^2(x) - \tan(x)}{x^2}$
   2. $B) \frac{\sec^2(x) - \tan(x)}{x^2}$
   3. $C) \frac{x\sec^2(x) + \tan(x)}{x^2}$
   4. $D) \frac{\sec^2(x)}{1}$