carney.kelly33
carney.kelly33 Jun 22, 2026 • 10 views

Solved examples of Product and Quotient Rules

Hey there! 👋 Learning about product and quotient rules can seem tricky at first, but with some practice, you'll nail it! Here's a quick study guide and a quiz to test your knowledge. Good luck! 🍀
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📚 Quick Study Guide

  • 🍎 Product Rule:
  • The derivative of a product of two functions, $u(x)$ and $v(x)$, is given by: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$.
  • ➕ In simpler terms: (first function's derivative × second function) + (first function × second function's derivative).
  • Quotient Rule:
  • The derivative of a quotient of two functions, $u(x)$ and $v(x)$, is given by: $\frac{d}{dx}[\frac{u(x)}{v(x)}] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.
  • ➖ In simpler terms: [(derivative of numerator × denominator) - (numerator × derivative of denominator)] / (denominator squared).
  • 💡 Tips:
  • Remember to apply the chain rule when dealing with composite functions.
  • 📝 Practice with various functions to get comfortable with these rules.

🧪 Practice Quiz

  1. What is the derivative of $f(x) = x^2 \sin(x)$?
    1. $2x \cos(x)$
    2. $x^2 \cos(x) + 2x \sin(x)$
    3. $2x \sin(x) - x^2 \cos(x)$
    4. $2x \sin(x)$

  2. Find the derivative of $f(x) = \frac{x}{x+1}$.
    1. $\frac{1}{(x+1)^2}$
    2. $\frac{2x+1}{(x+1)^2}$
    3. $\frac{1}{x+1}$
    4. $\frac{-1}{(x+1)^2}$

  3. Determine the derivative of $f(x) = (3x+2)(x^2-1)$.
    1. $9x^2 + 4x - 3$
    2. $3x^2 + 2x$
    3. $6x$
    4. $3x^2 + 4x - 3$

  4. Calculate the derivative of $f(x) = \frac{\cos(x)}{x}$.
    1. $\frac{-x\sin(x) - \cos(x)}{x^2}$
    2. $\frac{x\sin(x) - \cos(x)}{x^2}$
    3. $\frac{-\sin(x)}{1}$
    4. $\frac{\sin(x)}{x^2}$

  5. What is the derivative of $f(x) = (x^3 + 1)\tan(x)$?
    1. $(3x^2)\sec^2(x)$
    2. $(x^3 + 1)\sec^2(x) + 3x^2\tan(x)$
    3. $(x^3 + 1)\tan(x) + 3x^2\sec^2(x)$
    4. $(x^3 + 1)\sec^2(x) - 3x^2\tan(x)$

  6. Find the derivative of $f(x) = \frac{e^x}{x^2}$.
    1. $\frac{e^x(x-2)}{x^3}$
    2. $\frac{e^x(x^2-2x)}{x^4}$
    3. $\frac{e^x}{2x}$
    4. $\frac{e^x}{x^2}$

  7. Determine the derivative of $f(x) = (2x-1)\ln(x)$.
    1. $\frac{2}{x}$
    2. $2\ln(x) + \frac{2x-1}{x}$
    3. $\frac{2x-1}{x}$
    4. $2\ln(x) + \frac{1}{x}$
Click to see Answers
  1. B
  2. A
  3. A
  4. A
  5. B
  6. B
  7. B

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