Derivatives of Exponential Functions Formula Quick Guide

🚀 Derivatives of Exponential Functions: A Quick Guide

Exponential functions are a fundamental part of calculus, and understanding how to differentiate them is crucial. Here's a quick rundown of the key formulas:

Basic Exponential Function

The derivative of the basic exponential function $f(x) = e^x$ is itself:

\frac{d}{dx}(e^x) = e^x

General Exponential Function

For a general exponential function $f(x) = a^x$, where $a$ is a constant, the derivative is:

\frac{d}{dx}(a^x) = a^x \ln(a)

Exponential Function with Chain Rule

When dealing with a composite function $f(x) = e^{g(x)}$, you need to apply the chain rule:

\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)

Similarly, for $f(x) = a^{g(x)}$:

\frac{d}{dx}(a^{g(x)}) = a^{g(x)} \cdot g'(x) \cdot \ln(a)

📝 Examples

1. Example 1: Find the derivative of $f(x) = e^{3x}$.

   Using the chain rule:

   f'(x) = e^{3x} \cdot \frac{d}{dx}(3x) = e^{3x} \cdot 3 = 3e^{3x}

2. Example 2: Find the derivative of $f(x) = 2^x$.

   f'(x) = 2^x \ln(2)
3. Example 3: Find the derivative of $f(x) = 5^{x^2}$.

   Using the chain rule:

   f'(x) = 5^{x^2} \cdot \ln(5) \cdot \frac{d}{dx}(x^2) = 5^{x^2} \cdot \ln(5) \cdot 2x = 2x \cdot 5^{x^2} \ln(5)

💡 Key Takeaways

• The derivative of $e^x$ is always $e^x$.
• For $a^x$, remember to multiply by $\ln(a)$.
• Always apply the chain rule when the exponent is a function of $x$.