keith860
keith860 7d ago • 0 views

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

Hey everyone! 👋 Let's tackle differentiating $e^x$ and $a^x$ functions. It's easier than you think! 😉 This guide and quiz will help you master it!
🧮 Mathematics
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📚 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. $2x \cdot e^{2x}$
Click to see Answers
  1. B
  2. C
  3. B
  4. C
  5. B
  6. C
  7. B

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