Calculus Chain Rule Test Questions with Detailed Answers

📚 Quick Study Guide

• 🔗 Definition: The Chain Rule is used to find the derivative of a composite function.
• ➗ Formula: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$.
• 💡 Steps:
  1. Identify the outer function $f(u)$ and the inner function $g(x)$.
  2. Find the derivative of the outer function with respect to $u$, i.e., $\frac{df}{du}$.
  3. Find the derivative of the inner function with respect to $x$, i.e., $\frac{dg}{dx}$.
  4. Multiply the derivatives: $\frac{df}{du} \cdot \frac{dg}{dx}$.
  5. Substitute $g(x)$ back in for $u$.
• 📝 Common Functions:
  • Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
  • Trigonometric Functions: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$
  • Exponential Functions: $\frac{d}{dx}(e^x) = e^x$
  • Logarithmic Functions: $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• ⎰ Example: If $y = \sin(x^2)$, then $f(u) = \sin(u)$ and $g(x) = x^2$. Therefore, $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$.

🧪 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. $6x^2 + 6x + 3$
4. $3(2x + 1)$
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: Determine the derivative of $y = e^{5x^2}$.
1. $10xe^{5x^2}$
2. $e^{5x^2}$
3. $5x^2e^{5x^2-1}$
4. $10e^{5x^2}$
4. Question 4: Calculate the derivative of $y = \ln(x^2 + 1)$.
1. $\frac{2x}{x^2 + 1}$
2. $\frac{1}{x^2 + 1}$
3. $\frac{2x}{\ln(x^2 + 1)}$
4. $\frac{1}{2x}$
5. Question 5: What is the derivative of $y = \cos^2(x)$?
1. $-2\sin(x)\cos(x)$
2. $2\sin(x)\cos(x)$
3. $-\sin^2(x)$
4. $\sin(2x)$
6. Question 6: Find $\frac{dy}{dx}$ if $y = \sqrt{4x - 3}$.
1. $\frac{2}{\sqrt{4x - 3}}$
2. $\frac{1}{\sqrt{4x - 3}}$
3. $\frac{4}{\sqrt{4x - 3}}$
4. $\frac{2}{4x - 3}$
7. Question 7: Determine the derivative of $y = (x^3 + 2x)^4$.
1. $4(3x^2 + 2)(x^3 + 2x)^3$
2. $4(x^3 + 2x)^3$
3. $4(3x^2 + 2)$
4. $12x^2 + 8$