**UpStudy Free Solution:**

To find the derivative of \(\sqrt { x} \), we can rewrite the square root function in terms of an exponent. Specifically, \(\sqrt { x} \) can be written as \(x^ { 1/2} \).

Now, we can use the power rule for differentiation, which states that if \(f( x) = x^ n\), then \(f' ( x) = n x^ { n- 1} \).

Applying the power rule to \(x^ { 1/2} \):

\[\frac { d} { dx} \left ( x^ { 1/2} \right ) = \frac { 1} { 2} x^ { ( 1/2) - 1} = \frac { 1} { 2} x^ { - 1/2} \]

This can be further simplified to:

\[\frac { 1} { 2} x^ { - 1/2} = \frac { 1} { 2} \cdot \frac { 1} { \sqrt { x} } = \frac { 1} { 2\sqrt { x} } \]

Therefore, the derivative of \(\sqrt { x} \) is:

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

**Supplemental Knowledge**

To find the derivative of a function, we often use basic differentiation rules. For the function \(\sqrt { x} \), it is helpful to rewrite it in a form that makes differentiation straightforward using the power rule.

The power rule states that for any function \(f( x) = x^ n\), where \(n\) is a real number, the derivative is given by:

\[f' ( x) = nx^ { n- 1} \]

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