The derivative of the function \( e^{x-1} \) with respect to \( x \) can be found using the chain rule. The chain rule states that if you have a function of the form \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \).

Given \( f(x) = e^{x-1} \):

1. The outer function is \( e^u \), where \( u = x - 1 \).
2. The derivative of \( e^u \) with respect to \( u \) is \( e^u \).
3. The derivative of the inner function \( u = x - 1 \) with respect to \( x \) is 1.

Applying the chain rule:

\[
\frac{d}{dx} e^{x-1} = e^{x-1} \cdot 1 = e^{x-1}
\]

So, the derivative of \( e^{x-1} \) is \( e^{x-1} \).

Would you like more details on this process or have any questions?

Here are 5 related questions for further practice:

1. What is the derivative of \( e^{2x} \)?
2. How do you differentiate \( e^{x^2} \) using the chain rule?
3. What is the derivative of \( e^{3x+2} \)?
4. Find the derivative of \( e^{-x} \).
5. What is the derivative of \( e^{\sin(x)} \)?

**Tip:** When differentiating exponential functions with more complex exponents, always remember to apply the chain rule correctly by first differentiating the outer function and then multiplying by the derivative of the inner function.

Discover how to find the derivative of e^(x-1) using the chain rule with a detailed step-by-step explanation. This problem involves differentiating exponential functions and applying the chain rule effectively. Suitable for advanced high school students and beyond.

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