Solved examples: Differentiating e^x and a^x functions

📚 Quick Study Guide

• 🍎 The derivative of $e^x$ is simply $e^x$. That is, $\frac{d}{dx}(e^x) = e^x$.
• 💡 For $a^x$, where $a$ is a constant, the derivative is $a^x \cdot \ln(a)$. So, $\frac{d}{dx}(a^x) = a^x \ln(a)$.
• 📝 Remember that $\ln(e) = 1$. This is why the derivative of $e^x$ is just $e^x$, because it's like saying $e^x \cdot \ln(e) = e^x \cdot 1 = e^x$.
• 🧮 When dealing with more complex functions involving $e^x$ or $a^x$, remember to apply the chain rule.
• 📌 The chain rule states that $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

🧪 Practice Quiz

1. What is the derivative of $e^{3x}$ with respect to $x$?
1. $e^{3x}$
2. $3e^{3x}$
3. $e^3$
4. $3x \cdot e^{3x-1}$
2. What is the derivative of $5^x$ with respect to $x$?
1. $5^x$
2. $x \cdot 5^{x-1}$
3. $5^x \ln(5)$
4. $\ln(5)$
3. Find the derivative of $e^{x^2}$ with respect to $x$.
1. $e^{x^2}$
2. $2x \cdot e^{x^2}$
3. $2e^{x^2}$
4. $x^2 \cdot e^{x^2-1}$
4. What is the derivative of $2^{4x}$ with respect to $x$?
1. $2^{4x} \ln(2)$
2. $4 \cdot 2^{4x}$
3. $4 \cdot 2^{4x} \ln(2)$
4. $x \cdot 2^{4x-1}$
5. Calculate the derivative of $e^{-x}$ with respect to $x$.
1. $e^{-x}$
2. $-e^{-x}$
3. $x \cdot e^{-x-1}$
4. $-x \cdot e^{-x-1}$
6. Determine the derivative of $10^x$ with respect to $x$.
1. $10^x$
2. $x \cdot 10^{x-1}$
3. $10^x \ln(10)$
4. $\ln(10)$
7. What is the derivative of $e^{2x+1}$ with respect to $x$?
1. $e^{2x+1}$
2. $2e^{2x+1}$
3. $e^2$
4. $2x \cdot e^{2x}$