Test Questions on Applying Chain Rule within Quotient Rule

📚 Quick Study Guide

• 💡 Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.
• 🔗 Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.
• ➕ Combining the Rules: When applying the chain rule within the quotient rule, carefully identify the 'inner' and 'outer' functions.
• 📝 Derivative of Common Functions: Remember basic derivatives such as $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, and $(x^n)' = nx^{n-1}$.
• ✔️ Simplify: After applying the rules, simplify the expression as much as possible.

Practice Quiz

1. What is the derivative of $f(x) = \frac{\sin(2x)}{x}$?
• A) $\frac{2x\cos(2x) - \sin(2x)}{x^2}$
• B) $\frac{x\cos(2x) - \sin(2x)}{x^2}$
• C) $\frac{2\cos(2x)}{1}$
• D) $\frac{x\cos(2x) + \sin(2x)}{x^2}$
2. Find $f'(x)$ if $f(x) = \frac{(x^2 + 1)^3}{x}$.
• A) $\frac{(x^2 + 1)^2(5x^2 - 1)}{x^2}$
• B) $\frac{(x^2 + 1)^2(5x^2 + 1)}{x^2}$
• C) $\frac{3(x^2 + 1)^2}{1}$
• D) $\frac{(x^2 + 1)^3}{x^2}$
3. Determine the derivative of $f(x) = \frac{\cos(x^2)}{x^3}$.
• A) $\frac{-2x^4\sin(x^2) - 3x^2\cos(x^2)}{x^6}$
• B) $\frac{2x^4\sin(x^2) - 3x^2\cos(x^2)}{x^6}$
• C) $\frac{-\sin(x^2)}{3x^2}$
• D) $\frac{-2x\sin(x^2)}{3x^2}$
4. What is the derivative of $f(x) = \frac{\sqrt{x^2 + 1}}{x}$?
• A) $\frac{x^2 - 1}{x^2\sqrt{x^2 + 1}}$
• B) $\frac{-1}{x^2\sqrt{x^2 + 1}}$
• C) $\frac{1}{\sqrt{x^2 + 1}}$
• D) $\frac{-1}{x^2\sqrt{x^2 + 1}}$
5. Compute $f'(x)$ for $f(x) = \frac{\tan(3x)}{x^2}$.
• A) $\frac{3x^2\sec^2(3x) - 2x\tan(3x)}{x^4}$
• B) $\frac{3\sec^2(3x) - 2\tan(3x)}{x^3}$
• C) $\frac{\sec^2(3x)}{2x}$
• D) $\frac{3x^2\sec^2(3x) + 2x\tan(3x)}{x^4}$
6. Find the derivative of $f(x) = \frac{e^{2x}}{x + 1}$.
• A) $\frac{2(x+1)e^{2x} - e^{2x}}{(x+1)^2}$
• B) $\frac{e^{2x}}{1}$
• C) $\frac{2e^{2x}}{1}$
• D) $\frac{2e^{2x} - e^{2x}}{(x+1)^2}$
7. Calculate $f'(x)$ if $f(x) = \frac{\ln(x^2 + 1)}{x}$.
• A) $\frac{2x^2}{(x^2+1)^2}$
• B) $\frac{\frac{2x}{x^2 + 1} \cdot x - \ln(x^2 + 1)}{x^2}$
• C) $\frac{2x}{x^2 + 1}$
• D) $\frac{2x}{x^2}$