william.webster
william.webster Jan 27, 2026 β€’ 0 views

Complex Chain Rule Problems: Solved Examples for Students

Hey there! πŸ‘‹ Feeling a bit lost with complex chain rule problems? Don't worry, I got you! This guide breaks down the chain rule with easy-to-follow examples, plus a quiz to test your understanding. Let's get started! πŸš€
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cynthia464 17h ago

πŸ“š Quick Study Guide

  • πŸ”— The Chain Rule: Used to differentiate composite functions (functions within functions).
  • πŸ“ Formula: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$.
  • πŸ’‘ Steps: Identify the outer function $f(u)$ and the inner function $g(x)$. Find the derivatives of both, $f'(u)$ and $g'(x)$. Apply the chain rule formula.
  • βž• Example: To differentiate $\sin(x^2)$, let $u = x^2$. Then $\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$.

πŸ§ͺ Practice Quiz

  1. Question 1: What is the derivative of $y = (2x + 1)^3$?
    1. $6(2x + 1)^2$
    2. $3(2x + 1)^2$
    3. $6x + 3$
    4. $2(2x + 1)^2$
  2. Question 2: Find $\frac{dy}{dx}$ if $y = \sin(3x)$.
    1. $3\cos(3x)$
    2. $\cos(3x)$
    3. $-3\cos(3x)$
    4. $-\cos(3x)$
  3. Question 3: Differentiate $y = e^{5x}$.
    1. $5e^{5x}$
    2. $e^{5x}$
    3. $5x e^{5x-1}$
    4. $e^{x}$
  4. Question 4: What is the derivative of $y = \ln(4x)$?
    1. $\frac{1}{x}$
    2. $\frac{4}{x}$
    3. $\frac{1}{4x}$
    4. $4\ln(4)$
  5. Question 5: Find $\frac{dy}{dx}$ if $y = \sqrt{x^2 + 1}$.
    1. $\frac{x}{\sqrt{x^2 + 1}}$
    2. $\frac{1}{2\sqrt{x^2 + 1}}$
    3. $\frac{2x}{\sqrt{x^2 + 1}}$
    4. $\frac{1}{\sqrt{x^2 + 1}}$
  6. Question 6: Differentiate $y = (x^2 + 3x)^4$.
    1. $4(x^2 + 3x)^3(2x + 3)$
    2. $4(x^2 + 3x)^3$
    3. $(2x + 3)^4$
    4. $4(2x + 3)$
  7. Question 7: What is the derivative of $y = \cos^2(x)$?
    1. $-2\cos(x)\sin(x)$
    2. $2\cos(x)\sin(x)$
    3. $-\sin^2(x)$
    4. $\sin^2(x)$
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