Solved examples of Product and Quotient Rules

📚 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. $2\ln(x) + \frac{1}{x}$