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To determine the derivative of the function \( y = 5 + 7\sqrt{x} \), we will follow these steps:

1. **Differentiate the constant term** \( 5 \):
   The derivative of a constant is zero.

2. **Differentiate the term \( 7\sqrt{x} \)**:
   Rewrite \( \sqrt{x} \) as \( x^{1/2} \), so the term becomes \( 7x^{1/2} \).

3. **Apply the power rule**:
   The power rule states that \( \frac{d}{dx}x^n = nx^{n-1} \).

Differentiating \( 7x^{1/2} \) gives:
   \[
   \frac{d}{dx}(7x^{1/2}) = 7 \cdot \frac{1}{2}x^{-1/2} = \frac{7}{2\sqrt{x}}.
   \]

Thus, the derivative of the function is:
   \[
   \frac{dy}{dx} = \frac{7}{2\sqrt{x}}.
   \]

Would you like more details on any step, or do you have further questions?

### Related Questions:
1. What is the derivative of \( y = 3 + 8x^{1/3} \)?
2. How would you differentiate \( y = 5x^{-1/2} + 7 \)?
3. What is the second derivative of \( y = 7\sqrt{x} \)?
4. How does the power rule apply to negative exponents?
5. What are the derivatives of trigonometric functions?

### Tip:
When differentiating a function, always simplify the terms as much as possible before applying the differentiation rules. This often makes the process easier.

Determine the derivative of the following function: y = 5 + 7√x

Learn how to differentiate y = 5 + 7√x by applying the power rule. Step-by-step explanation provided for students in Grades 11-12. Explore key differentiation concepts, including the power rule, to find the derivative of this function.

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