# derivative of square root x

#### Understand the Problem

The question is asking for the derivative of the function square root of x. To solve this, we will apply the rules of differentiation, specifically the power rule, since the square root can be expressed as x raised to the power of 1/2.

$$\frac{1}{2 \sqrt{x}}$$

The final answer is ( \frac{1}{2 \sqrt{x}} )

#### Steps to Solve

1. Rewrite the square root function using exponentiation

The square root of $x$ can be expressed as an exponent:

$$\sqrt{x} = x^{1/2}$$

2. Apply the power rule for differentiation

The power rule states that the derivative of $x^n$ is $nx^{n-1}$.

For our function $x^{1/2}$, apply the power rule:

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

3. Simplify the exponent

Simplify the exponent $\frac{1}{2} - 1$:

$$\frac{1}{2} - 1 = -\frac{1}{2}$$

So, the expression becomes:

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

4. Rewrite to standard form

Rewrite $x^{-1/2}$ as $\frac{1}{\sqrt{x}}$:

$$\frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} \cdot \frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{x}}$$

The final answer is ( \frac{1}{2 \sqrt{x}} )