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.