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.