To find the derivative of the function \( f(x) = \sqrt{x} \), you can rewrite the square root function in exponential form:

\[
f(x) = x^{\frac{1}{2}}
\]

Now, use the power rule for differentiation, which states:

\[
\frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}
\]

Here, \( n = \frac{1}{2} \). Applying the power rule:

\[
f'(x) = \frac{1}{2} \cdot x^{\frac{1}{2} - 1} = \frac{1}{2} \cdot x^{-\frac{1}{2}}
\]

This simplifies to:

\[
f'(x) = \frac{1}{2\sqrt{x}}
\]

So, the derivative of \( f(x) = \sqrt{x} \) is:

\[
f'(x) = \frac{1}{2\sqrt{x}}
\]

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you find the derivative of \( f(x) = \sqrt[3]{x} \)?
2. What is the derivative of \( f(x) = x^n \) for any value of \( n \)?
3. Can you apply the chain rule to find the derivative of \( f(x) = \sqrt{2x+3} \)?
4. How would you find the second derivative of \( f(x) = \sqrt{x} \)?
5. How does the derivative of \( f(x) = \sqrt{x} \) change if \( x \) is negative?

**Tip:** When dealing with square roots in differentiation, it's often helpful to rewrite them in exponential form, as it simplifies the application of the power rule.

Learn how to find the derivative of the function f(x) = √x step-by-step. This problem covers differentiation using the power rule, suitable for advanced high school students.

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