robin.carr
robin.carr 6d ago โ€ข 0 views

Generalized Power Rule Solved Examples with Detailed Steps

Hey there! ๐Ÿ‘‹ Feeling a bit lost with the Generalized Power Rule in calculus? Don't worry, it can be tricky! This study guide + quiz will help you master it with easy examples and practice questions. Let's ace this together! ๐Ÿ’ช
๐Ÿงฎ Mathematics

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shawn444 Dec 31, 2025

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข The Generalized Power Rule states that if $y = [f(x)]^n$, then $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$.
  • ๐Ÿ“ In simpler terms: Bring down the power, keep the inside function the same, decrease the power by one, and multiply by the derivative of the inside function.
  • ๐Ÿ’ก Remember the chain rule is essential here! The 'inside' function's derivative is crucial.
  • ๐Ÿ“ˆ It's widely used for differentiating composite functions (functions within functions).
  • ๐Ÿงช Practice with various functions like trigonometric, exponential, and algebraic functions inside the power.
  • ๐Ÿ”‘ Always simplify your answer to its simplest form where possible.

๐Ÿง  Practice Quiz

  1. What is the derivative of $y = (2x + 1)^3$?
    1. $6(2x + 1)^2$
    2. $3(2x + 1)^2$
    3. $6x + 3$
    4. $3(2x + 1)^2 \cdot 2$
  2. Find $\frac{dy}{dx}$ if $y = \sqrt{3x^2 + 5}$?
    1. $\frac{3x}{\sqrt{3x^2 + 5}}$
    2. $\frac{1}{2\sqrt{3x^2 + 5}}$
    3. $6x$
    4. $\frac{3x}{2\sqrt{3x^2 + 5}}$
  3. Differentiate $y = \sin^2(x)$ with respect to $x$.
    1. $2\sin(x)$
    2. $\cos^2(x)$
    3. $2\sin(x)\cos(x)$
    4. $\sin(x)\cos(x)$
  4. What is the derivative of $y = \cos(5x)^4$?
    1. $-4\cos^3(5x)\sin(5x)$
    2. $-20\cos^3(5x)\sin(5x)$
    3. $4\cos^3(5x)$
    4. $-4\sin(5x)^3$
  5. Find $\frac{dy}{dx}$ for $y = (x^3 - 2x + 1)^5$.
    1. $5(3x^2 - 2)^4$
    2. $5(x^3 - 2x + 1)^4$
    3. $5(x^3 - 2x + 1)^4 (3x^2 - 2)$
    4. $5(x^3 - 2x + 1)^4 (x^3 - 2x + 1)$
  6. Calculate the derivative of $y = e^{(x^2)}$?
    1. $e^{(x^2)}$
    2. $2xe^{(x^2)}$
    3. $e^{(2x)}$
    4. $2x$
  7. What is the derivative of $y = \tan(x^2 + 1)$?
    1. $\sec^2(x^2 + 1)$
    2. $2x\sec^2(x^2 + 1)$
    3. $\sec^2(2x)$
    4. $2x\tan(x^2 + 1)$
Click to see Answers
  1. D
  2. A
  3. C
  4. B
  5. C
  6. B
  7. B

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