Step-by-step examples of the power of a product rule.

📚 Quick Study Guide

• 🍎 The Product Rule: Used to differentiate functions that are the product of two or more functions.
• 📐 Formula: If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$.
• 💡 Tip: Remember to identify $u(x)$ and $v(x)$ correctly before applying the formula.
• ✍️ Notation: $u'(x)$ and $v'(x)$ represent the derivatives of $u(x)$ and $v(x)$ with respect to $x$.
• ➕ Application: Break down complex functions into simpler components to make differentiation easier.
• 🧠 Example: To differentiate $x^2 \sin(x)$, let $u(x) = x^2$ and $v(x) = \sin(x)$.
• 🔑 Key Steps: Differentiate $u(x)$ and $v(x)$ separately, then apply the product rule formula.

🧪 Practice Quiz

1. What is the product rule used for?
   1. A) Integrating the product of two functions
   2. B) Differentiating the quotient of two functions
   3. C) Differentiating the product of two functions
   4. D) Finding the limit of a product
2. Given $y = (x^3 + 1)\cos(x)$, what is $u(x)$ and $v(x)$?
   1. A) $u(x) = \cos(x)$, $v(x) = x^3$
   2. B) $u(x) = x^3 + 1$, $v(x) = \cos(x)$
   3. C) $u(x) = x^3$, $v(x) = 1$
   4. D) $u(x) = 1$, $v(x) = \cos(x)$
3. What is the derivative of $y = x \cdot e^x$?
   1. A) $e^x$
   2. B) $x \cdot e^x$
   3. C) $e^x + x \cdot e^x$
   4. D) $x + e^x$
4. If $f(x) = (2x + 3)(x^2 - 1)$, what is $f'(x)$?
   1. A) $2x + 3$
   2. B) $x^2 - 1$
   3. C) $6x^2 + 6x - 2$
   4. D) $4x - 2$
5. What is the derivative of $y = \sqrt{x} \sin(x)$?
   1. A) $\frac{\cos(x)}{2\sqrt{x}}$
   2. B) $\frac{\sin(x)}{2\sqrt{x}} + \sqrt{x} \cos(x)$
   3. C) $\frac{\sin(x)}{2\sqrt{x}} - \sqrt{x} \cos(x)$
   4. D) $\frac{\cos(x)}{2\sqrt{x}} + \sqrt{x} \sin(x)$
6. Given $y = x^2 \ln(x)$, what is $\frac{dy}{dx}$?
   1. A) $2x \ln(x)$
   2. B) $\frac{1}{x}$
   3. C) $2x + \frac{1}{x}$
   4. D) $2x \ln(x) + x$
7. What is the correct formula for the product rule?
   1. A) $\frac{dy}{dx} = u(x)v(x)$
   2. B) $\frac{dy}{dx} = u'(x)v'(x)$
   3. C) $\frac{dy}{dx} = u'(x)v(x) - u(x)v'(x)$
   4. D) $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$