The derivative of $e^x$ with respect to $x$ is $e^x$ itself, which is a unique property of the exponential function. Here are a few examples illustrating how this property is used in differentiation:

1. Example 1: Basic Derivative $\frac{d}{dx} e^x = e^x$ Here, since $e^x$ is the function, its derivative is also $e^x$.

2. Example 2: Exponential Function with a Constant Multiplier $\frac{d}{dx} (5e^x) = 5 \cdot \frac{d}{dx} (e^x) = 5e^x$ When $e^x$ is multiplied by a constant (5 in this case), the constant is carried through the derivative, giving $5e^x$.

3. Example 3: Exponential Function with a Constant Power $e^{3x}$ $\frac{d}{dx} (e^{3x}) = 3e^{3x}$ By the chain rule, since $e^{3x}$ has an inner function $3x$, we take the derivative of $3x$, which is 3, and multiply it by $e^{3x}$.

4. Example 4: Exponential Function as Part of a Sum $\frac{d}{dx} (e^x + x^2) = e^x + 2x$ Here, we differentiate each term separately. The derivative of $e^x$ is $e^x$, and the derivative of $x^2$ is $2x$.

5. Example 5: Exponential Function in a Product $\frac{d}{dx} (x \cdot e^x) = x \cdot \frac{d}{dx} (e^x) + e^x \cdot \frac{d}{dx} (x) = x \cdot e^x + e^x = (x+1)e^x$ Using the product rule, we find the derivative of $x \cdot e^x$, which involves differentiating both $x$ and $e^x$ and summing their contributions.

Would you like further clarification on these examples or any other topic related to $e^x$ differentiation?

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Related Questions:

1. How is the chain rule applied to more complex functions involving $e^x$?
2. Can we differentiate other exponential functions, like $a^x$ where $a \neq e$?
3. What happens when differentiating $e^x$ in the context of an integral?
4. How do exponential functions behave when combined with trigonometric functions?
5. Are there any interesting applications of $e^x$ in real-world problems?

Tip: When dealing with exponential functions with different powers or bases, remember the chain rule and the natural logarithm