We are tasked with differentiating the function:

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

Step-by-step Solution:

1. Rewrite the square root in exponential form: $y = (e^x + 14)^{1/2}.$

2. Use the chain rule: The derivative of $y = u^{1/2}$ is: $\frac{dy}{dx} = \frac{1}{2}u^{-1/2} \cdot \frac{du}{dx}.$

   Here, $u = e^x + 14$.

3. Differentiate $u = e^x + 14$: The derivative of $e^x$ is $e^x$, and the derivative of 14 is 0. Thus: $\frac{du}{dx} = e^x.$

4. Substitute $u$ and $\frac{du}{dx}$ into the formula: $\frac{dy}{dx} = \frac{1}{2}(e^x + 14)^{-1/2} \cdot e^x.$

5. Simplify the result: $\frac{dy}{dx} = \frac{e^x}{2\sqrt{e^x + 14}}.$

Final Answer:

$\frac{dy}{dx} = \frac{e^x}{2\sqrt{e^x + 14}}.$

Would you like a detailed explanation of any step or additional examples?

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Related Questions:

1. How do you apply the chain rule to composite functions?
2. What is the derivative of $\sqrt{x + c}$?
3. How does the exponential function $e^x$ behave during differentiation?
4. What happens to constants like 14 when differentiated?
5. How do we simplify derivatives involving square roots?

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Tip:

Always check whether the function can be rewritten in simpler forms (e.g., square roots to fractional exponents) to make differentiation more straightforward!