Test Questions for Differentiating Mixed Rule Calculus Problems

📚 Quick Study Guide

• 🔍 Chain Rule: Use when differentiating composite functions. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
• 💡 Product Rule: Use when differentiating the product of two functions. If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$.
• 📝 Quotient Rule: Use when differentiating the quotient of two functions. If $y = \frac{u(x)}{v(x)}$, then $\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.
• ➕ Sum/Difference Rule: The derivative of a sum or difference is the sum or difference of the derivatives. $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$.
• 🧮 Power Rule: If $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.
• 🌱 Exponential Rule: If $y = e^x$, then $\frac{dy}{dx} = e^x$. If $y = a^x$, then $\frac{dy}{dx} = a^x \ln(a)$.
• 📈 Logarithmic Rule: If $y = \ln(x)$, then $\frac{dy}{dx} = \frac{1}{x}$. If $y = \log_a(x)$, then $\frac{dy}{dx} = \frac{1}{x \ln(a)}$.

Practice Quiz

1. What rule should you use FIRST to differentiate $y = (x^2 + 1)e^x$?
   1. Product Rule
   2. Quotient Rule
   3. Chain Rule
   4. Power Rule
2. Find the derivative of $y = \sin(x^2)$:
   1. $2x\cos(x^2)$
   2. $\cos(x^2)$
   3. $-2x\cos(x^2)$
   4. $2\cos(x)$
3. What is the derivative of $y = \frac{x^3}{\cos(x)}$?
   1. $\frac{3x^2\cos(x) + x^3\sin(x)}{\cos^2(x)}$
   2. $\frac{3x^2\cos(x) - x^3\sin(x)}{\cos^2(x)}$
   3. $\frac{3x^2}{\sin(x)}$
   4. $\frac{x^2}{\cos^2(x)}$
4. Differentiate $y = \ln(5x)$:
   1. $\frac{1}{x}$
   2. $\frac{1}{5x}$
   3. $5$
   4. $\frac{5}{x}$
5. What rule should you use FIRST to differentiate $y = \sqrt{x^3 + 1}$?
   1. Chain Rule
   2. Product Rule
   3. Quotient Rule
   4. Power Rule
6. Find the derivative of $y = x^2 \tan(x)$:
   1. $2x \tan(x) + x^2 \sec^2(x)$
   2. $2x \sec^2(x)$
   3. $2x \tan(x) + \sec^2(x)$
   4. $x^2 \sec^2(x)$
7. Differentiate $y = e^{x^2 + 1}$:
   1. $2xe^{x^2 + 1}$
   2. $e^{x^2 + 1}$
   3. $2xe^x$
   4. $e^{2x}$