Hard Chain Rule examples

📚 Quick Study Guide

🔍 The Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$. This means you take the derivative of the outer function with respect to the inner function, and then multiply by the derivative of the inner function with respect to $x$.

💡 Steps to Apply:
• 1. Identify the outer function $f(u)$ and the inner function $g(x)$.
• 2. Find the derivative of the outer function, $f'(u)$.
• 3. Find the derivative of the inner function, $g'(x)$.
• 4. Apply the chain rule formula: $f'(g(x)) \cdot g'(x)$.

📝 Common Derivatives:
• $\frac{d}{dx} [x^n] = nx^{n-1}$
• $\frac{d}{dx} [\sin(x)] = \cos(x)$
• $\frac{d}{dx} [\cos(x)] = -\sin(x)$
• $\frac{d}{dx} [e^x] = e^x$

Practice Quiz

1. What is the derivative of $y = (2x + 1)^3$?
1. A) $6(2x + 1)^2$
2. B) $3(2x + 1)^2$
3. C) $6x^2 + 6x + 3$
4. D) $3(2x + 1)^2 \cdot 2$
2. Find $\frac{dy}{dx}$ if $y = \sin(x^2)$.
1. A) $\cos(x^2)$
2. B) $2x\cos(x^2)$
3. C) $-\cos(x^2)$
4. D) $2x\sin(x^2)$
3. What is the derivative of $y = e^{5x}$?
1. A) $e^{5x}$
2. B) $5e^{5x}$
3. C) $e^5$
4. D) $5e^{x}$
4. Calculate $\frac{dy}{dx}$ for $y = \cos(3x + 2)$.
1. A) $-\sin(3x + 2)$
2. B) $3\sin(3x + 2)$
3. C) $-3\sin(3x + 2)$
4. D) $\sin(3x + 2)$
5. Determine the derivative of $y = \sqrt{4x - 1}$.
1. A) $\frac{2}{\sqrt{4x - 1}}$
2. B) $\frac{1}{2\sqrt{4x - 1}}$
3. C) $\frac{4}{\sqrt{4x - 1}}$
4. D) $\frac{2}{\sqrt{x}}$
6. If $y = (x^2 + 3x)^4$, find $\frac{dy}{dx}$.
1. A) $4(x^2 + 3x)^3$
2. B) $4(2x + 3)^3$
3. C) $4(x^2 + 3x)^3 (2x + 3)$
4. D) $(2x + 3)$
7. What is the derivative of $y = \sin^2(x)$?
1. A) $\cos^2(x)$
2. B) $2\sin(x)$
3. C) $2\sin(x)\cos(x)$
4. D) $2\cos(x)$