Chain Rule Examples: Solved Derivative Problems with Explanations

📚 Quick Study Guide

🔍 The chain rule is used to find the derivative of a composite function. It states that if $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$. 💡 To apply the chain rule, identify the outer function $f(u)$ and the inner function $g(x)$. 📝 Find the derivative of the outer function with respect to the inner function, $f'(u)$. 🧮 Find the derivative of the inner function with respect to $x$, $g'(x)$. 📌 Multiply the two derivatives together: $f'(g(x)) \cdot g'(x)$. ➗ Remember to simplify the resulting expression.

Practice Quiz

1. What is the derivative of $y = (2x + 1)^3$ with respect to $x$?
   1. $6(2x + 1)^2$
   2. $3(2x + 1)^2$
   3. $6x + 3$
   4. $24x^2 + 24x + 6$
2. Find $\frac{dy}{dx}$ if $y = \sin(x^2)$.
   1. $\cos(x^2)$
   2. $2x\cos(x^2)$
   3. $-\cos(x^2)$
   4. $-2x\cos(x^2)$
3. Determine the derivative of $y = e^{5x}$.
   1. $e^{5x}$
   2. $5e^{5x}$
   3. $e^5$
   4. $5x e^{5x-1}$
4. Calculate $\frac{d}{dx} \cos(3x + 2)$.
   1. $-\sin(3x + 2)$
   2. $3\sin(3x + 2)$
   3. $-3\sin(3x + 2)$
   4. $\sin(3x + 2)$
5. Differentiate $y = \sqrt{4x - 1}$.
   1. $\frac{1}{\sqrt{4x - 1}}$
   2. $\frac{2}{\sqrt{4x - 1}}$
   3. $\frac{4}{\sqrt{4x - 1}}$
   4. $\frac{1}{2\sqrt{4x - 1}}$
6. What is the derivative of $y = \ln(x^3 + 1)$?
   1. $\frac{1}{x^3 + 1}$
   2. $\frac{3x^2}{x^3 + 1}$
   3. $\frac{3x^2}{\ln(x^3 + 1)}$
   4. $\frac{1}{3x^2}$
7. Find $\frac{dy}{dx}$ for $y = (x^2 + 3x)^4$.
   1. $4(x^2 + 3x)^3$
   2. $(2x + 3)(x^2 + 3x)^3$
   3. $4(2x + 3)(x^2 + 3x)^3$
   4. $4(x^2 + 3x)^3 (x^2+3x)$