To differentiate $\sqrt{e^x}$ with respect to $x$, we will use the chain rule. The function can be rewritten as $(e^x)^{1/2}$.

Let's go through the steps:

1. Rewrite the function: $\sqrt{e^x} = (e^x)^{1/2}$

2. Differentiate using the chain rule: The chain rule states that if you have a composite function $f(g(x))$, then $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.

   Let $u = e^x$. Then $\sqrt{e^x} = u^{1/2}$.

   • Differentiate $u^{1/2}$ with respect to $u$: $\frac{d}{du} u^{1/2} = \frac{1}{2} u^{-1/2}$

   • Differentiate $e^x$ with respect to $x$: $\frac{d}{dx} e^x = e^x$

3. Apply the chain rule: $\frac{d}{dx} \sqrt{e^x} = \frac{d}{du} u^{1/2} \cdot \frac{du}{dx} = \frac{1}{2} u^{-1/2} \cdot e^x$

4. Substitute back $u = e^x$: $\frac{1}{2} (e^x)^{-1/2} \cdot e^x = \frac{1}{2} \cdot \frac{e^x}{\sqrt{e^x}} = \frac{1}{2} \cdot \sqrt{e^x}$

Therefore, the derivative of $\sqrt{e^x}$ with respect to $x$ is: $\frac{d}{dx} \sqrt{e^x} = \frac{1}{2} \sqrt{e^x}$

Would you like more details or have any questions?

Here are 5 related questions you might want to ask:

1. How do you apply the chain rule to more complex functions?
2. What are some common mistakes to avoid when using the chain rule?
3. Can you explain the product rule for differentiation?
4. How do you differentiate functions involving natural logarithms?
5. What is the process for differentiating trigonometric functions?

Tip: Always remember to rewrite functions in a simpler form, if possible, before differentiating. This can make applying rules like the chain rule much easier.