Complex Chain Rule Problems: Solved Examples for Students

📚 Quick Study Guide

• 🔗 The Chain Rule: Used to differentiate composite functions (functions within functions).
• 📝 Formula: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$.
• 💡 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.
• ➕ Example: To differentiate $\sin(x^2)$, let $u = x^2$. Then $\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$.

🧪 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. $\sin^2(x)$