scott904
scott904 1d ago β€’ 0 views

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

Hey everyone! πŸ‘‹ Trying to wrap your head around the product rule in calculus? It can seem tricky at first, but with some clear examples, you'll get the hang of it in no time! Let's break it down with a quick study guide and then test your knowledge with a practice quiz. Good luck!πŸ€
🧠 General Knowledge

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susan_lynch Dec 27, 2025

πŸ“š Quick Study Guide

  • 🍎 The Product Rule: Used to differentiate functions that are the product of two or more functions.
  • πŸ“ Formula: If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$.
  • πŸ’‘ 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 $x$.
  • βž• 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 $v(x) = \sin(x)$.
  • πŸ”‘ Key Steps: Differentiate $u(x)$ and $v(x)$ separately, then apply the product rule formula.

πŸ§ͺ 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) $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$
Click to see Answers
  1. C
  2. B
  3. C
  4. C
  5. B
  6. D
  7. D

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