To find the derivative of $e^{x-1}$ without explicitly using the chain rule, you can proceed as follows:

Given the function $f(x) = e^{x-1}$, we know that:

$f(x) = e^{x-1} = \frac{e^x}{e}$

Since $e^{x-1}$ can be rewritten as $\frac{e^x}{e}$, the function simplifies to:

$f(x) = \frac{1}{e} \cdot e^x$

Here, $\frac{1}{e}$ is just a constant, and the derivative of $e^x$ with respect to $x$ is $e^x$. Therefore, the derivative is:

$f'(x) = \frac{1}{e} \cdot e^x = e^{x-1}$

So, even without directly using the chain rule, the derivative of $e^{x-1}$ is still $e^{x-1}$.

Would you like to explore this method further, or do you have any other questions?

Here are 5 related questions for further practice:

1. Differentiate $\frac{e^x}{2}$ with respect to $x$.
2. Find the derivative of $e^{2x+3}$ using simplification techniques.
3. How do you differentiate $e^{x^2 - 1}$ without the chain rule?
4. Differentiate $\frac{e^{2x}}{e^2}$ with respect to $x$.
5. What is the derivative of $e^x$ if rewritten as $e \times e^{x-1}$?

Tip: Recognizing when a function can be rewritten in a simpler form, especially when constants are involved, can help in differentiating without explicitly applying the chain rule.