To find the derivative of the function \( f(x) = \sqrt{x} - 1 \), we'll differentiate each term separately.

1. **Rewrite the function**: 
   \[
   f(x) = x^{1/2} - 1
   \]

2. **Differentiate term by term**:
   - The derivative of \( x^{1/2} \) with respect to \( x \) is:
     \[
     \frac{d}{dx}\left( x^{1/2} \right) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}
     \]
   - The derivative of a constant (−1) is 0.

3. **Combine the results**:
   \[
   f'(x) = \frac{1}{2\sqrt{x}} - 0 = \frac{1}{2\sqrt{x}}
   \]

So, the derivative of \( \sqrt{x} - 1 \) is \( \frac{1}{2\sqrt{x}} \).

Would you like more details or have any questions?

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Here are 8 related questions to further explore this topic:

1. How would you find the derivative of \( \sqrt{x} + 3x^2 \)?
2. What is the derivative of \( \sqrt{x^2 + 1} \)?
3. How do you differentiate the function \( \sqrt{x} \) using the chain rule?
4. What is the derivative of \( \frac{1}{\sqrt{x}} \)?
5. How do you find the derivative of a square root function in general?
6. What is the second derivative of \( \sqrt{x} - 1 \)?
7. How would you differentiate \( x^{3/2} - x^{1/2} \)?
8. How do you apply the product rule to functions involving square roots?

**Tip:** When differentiating functions involving square roots, it can be helpful to rewrite the square root as a fractional exponent before applying the power rule.

Discover how to find the derivative of √x - 1 with a clear step-by-step solution. Learn the differentiation of square root functions and apply it to solve similar problems. Suitable for Grades 11-12.

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