Generalized Power Rule Solved Examples with Detailed Steps

📚 Quick Study Guide

• 🔢 The Generalized Power Rule states that if $y = [f(x)]^n$, then $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$.
• 📝 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.
• 💡 Remember the chain rule is essential here! The 'inside' function's derivative is crucial.
• 📈 It's widely used for differentiating composite functions (functions within functions).
• 🧪 Practice with various functions like trigonometric, exponential, and algebraic functions inside the power.
• 🔑 Always simplify your answer to its simplest form where possible.

🧠 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. $2x\tan(x^2 + 1)$