We need to differentiate the given function:

$y = \sqrt{e^x + 14}$

Step 1: Apply the chain rule

The square root can be rewritten as a power: $y = (e^x + 14)^{\frac{1}{2}}$

Using the chain rule: $\frac{dy}{dx} = \frac{1}{2}(e^x + 14)^{-\frac{1}{2}} \cdot \frac{d}{dx}(e^x + 14)$

Step 2: Differentiate the inner function

The derivative of $e^x + 14$ is: $\frac{d}{dx}(e^x + 14) = e^x$

Step 3: Combine the results

Substitute $\frac{d}{dx}(e^x + 14)$ into the expression: $\frac{dy}{dx} = \frac{1}{2}(e^x + 14)^{-\frac{1}{2}} \cdot e^x$

Step 4: Simplify the expression

The final derivative is: $\frac{dy}{dx} = \frac{e^x}{2\sqrt{e^x + 14}}$

Let me know if you'd like further clarification or a detailed breakdown of any step.

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Here are some follow-up questions you might explore:

1. How would the differentiation change if $14$ was replaced with another function of $x$?
2. Can you integrate $\frac{e^x}{2\sqrt{e^x + 14}}$ back to verify the original function?
3. What is the behavior of $\frac{dy}{dx}$ as $x \to \infty$?
4. What is the second derivative of $y = \sqrt{e^x + 14}$?
5. How does the derivative change if the square root is replaced with a cube root?

Tip: Always rewrite roots and radicals as fractional exponents when applying differentiation rules—it simplifies the process!