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The given function is:

\[
G(x) = \sqrt{x} + 4e^x
\]

We need to find the derivative \( G'(x) \).

### Step-by-step solution:

1. **Differentiate** \( \sqrt{x} \):
   \[
   \frac{d}{dx} \left( \sqrt{x} \right) = \frac{1}{2\sqrt{x}}
   \]

2. **Differentiate** \( 4e^x \):
   \[
   \frac{d}{dx} \left( 4e^x \right) = 4e^x
   \]

### Now, sum the two results:
\[
G'(x) = \frac{1}{2\sqrt{x}} + 4e^x
\]

Thus, the derivative of the function is:

\[
G'(x) = \frac{1}{2\sqrt{x}} + 4e^x
\]

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How do you apply the chain rule when differentiating composite functions?
2. What is the derivative of \( e^x \) when it's multiplied by a constant?
3. How do you find the derivative of other square root functions like \( \sqrt{ax + b} \)?
4. Can you explain how to differentiate logarithmic functions?
5. How would you compute the second derivative of the given function \( G(x) \)?

**Tip**: When differentiating a function that contains constants, treat them as if they are unaffected by the differentiation.

Differentiate the function G(x) = √x + 4e^x. Find G'(x).

Learn how to differentiate the function G(x) = √x + 4e^x. This problem explains key differentiation rules, including the derivative of square roots and exponential functions. Ideal for students in advanced high school calculus or early college calculus courses.

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