Given the function f, of, x, equals, e, start superscript, x, minus, 1, end superscript, commaf(x)=e 
x−1
 , write an expression that represents the derivative of ff using the limit shown below. You do not have to simplify your answer.

To find the derivative of the function \( f(x) = e^{x - 1} \) using the limit definition of a derivative, we start by recalling the definition:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

Given that \( f(x) = e^{x - 1} \), we will apply this definition step-by-step:

1. **Substitute \( f(x) \) into the limit definition**:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

2. **Find \( f(x + h) \)**: 

Since \( f(x) = e^{x - 1} \), then

\[
f(x + h) = e^{(x + h) - 1} = e^{x + h - 1}
\]

3. **Substitute \( f(x + h) \) and \( f(x) \) into the limit expression**:

\[
f'(x) = \lim_{h \to 0} \frac{e^{x + h - 1} - e^{x - 1}}{h}
\]

This is the expression for the derivative of \( f(x) = e^{x - 1} \) using the limit definition of the derivative. 

If you need further details or have any questions, feel free to ask!

### Related Questions:
1. How would you simplify this expression further?
2. What is the value of \( f'(x) \) when \( x = 0 \)?
3. What general rule for the derivative of exponential functions does this expression relate to?
4. How does the presence of \( x-1 \) in the exponent affect the derivative?
5. What are the steps to compute limits involving exponential functions?

### Tip:
When using the limit definition of a derivative, breaking down each step carefully will ensure you correctly substitute and simplify expressions.

Given the function f(x) = e^(x - 1), write an expression that represents the derivative of f using the limit shown below. You do not have to simplify your answer.

Learn how to find the derivative of the exponential function f(x) = e^(x - 1) using the limit definition of a derivative. This step-by-step solution explores key calculus concepts, including limits and derivatives, and is ideal for students in Grades 11-12.

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