Calculus Test Questions: Differentiating using the Product and Chain Rule.

📚 Quick Study Guide

• ⚛️ Product Rule: Used when differentiating the product of two functions. If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$.
• 🔗 Chain Rule: Used when differentiating a composite function. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
• 💡 Combining the Rules: Sometimes, you'll need to use both the product and chain rule in the same problem! Identify the 'outer' and 'inner' functions carefully.
• 📝 Derivative of Common Functions: Remember basic derivatives like $\frac{d}{dx}(x^n) = nx^{n-1}$, $\frac{d}{dx}(\sin x) = \cos x$, and $\frac{d}{dx}(\cos x) = -\sin x$.

Practice Quiz

1. Question 1: What is the derivative of $y = x^2 \sin(x)$?
   1. A) $2x \cos(x)$
   2. B) $x^2 \cos(x) + 2x \sin(x)$
   3. C) $2x \sin(x) - x^2 \cos(x)$
   4. D) $2x \sin(x)$
2. Question 2: What is the derivative of $y = (2x + 1)^3$?
   1. A) $3(2x + 1)^2$
   2. B) $6(2x + 1)^2$
   3. C) $3(2x + 1)^2 \cdot 2$
   4. D) $6x^2 + 6x + 3$
3. Question 3: What is the derivative of $y = \cos(x^2)$?
   1. A) $-\sin(x^2)$
   2. B) $-2x \sin(x^2)$
   3. C) $2x \sin(x^2)$
   4. D) $\sin(x^2)$
4. Question 4: What is the derivative of $y = x \sqrt{x+1}$?
   1. A) $\frac{3x+2}{2\sqrt{x+1}}$
   2. B) $\frac{1}{2\sqrt{x+1}}$
   3. C) $\frac{x}{2\sqrt{x+1}}$
   4. D) $\frac{x+2}{2\sqrt{x+1}}$
5. Question 5: What is the derivative of $y = \sin(2x) \cos(x)$?
   1. A) $2\cos(2x) \sin(x) - \sin(2x) \sin(x)$
   2. B) $2\cos(2x) \cos(x) - \sin(2x) \sin(x)$
   3. C) $\cos(2x) \cos(x) - \sin(2x) \sin(x)$
   4. D) $2\cos(2x) \cos(x) + \sin(2x) \sin(x)$
6. Question 6: What is the derivative of $y = \tan(x^3)$?
   1. A) $\sec^2(x^3)$
   2. B) $3x^2 \sec^2(x^3)$
   3. C) $3x^2 \tan(x^3) \sec(x^3)$
   4. D) $3x^2 \sec^2(x)$
7. Question 7: What is the derivative of $y = \frac{x}{\sqrt{x^2 + 1}}$?
   1. A) $\frac{1}{(x^2 + 1)^{3/2}}$
   2. B) $\frac{1}{\sqrt{x^2 + 1}}$
   3. C) $\frac{-1}{(x^2 + 1)^{3/2}}$
   4. D) $\frac{x}{(x^2 + 1)^{3/2}}$