Derivative of the Square Root sqrt(x) - Proof and Explanation

Proof

1. Definition of $\sqrt{x}$:

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

2. Using the power rule:

   $$\text{If }y=x^n,\text{ then }{y}^{\prime }=n{x}^{n-1}$$

   Let $n=\frac{1}{2}$:

   $$\frac{\text{d}}{\text{d}x}{x}^{\frac{1}{2}}=\frac{1}{2}{x}^{\frac{1}{2}-1}$$

3. Simplify the expression:

   $$\frac{\text{d}}{\text{d}x}{x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}$$

4. Rewrite using the square root:

   $$x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}$$

Thus, the derivative of $\sqrt{x}$ is:

Explanation

To understand the derivative of $\sqrt{x}$, we start by expressing $\sqrt{x}$ as a power of $x$. Specifically, $\sqrt{x}$ can be written as $x^{\frac{1}{2}}$. This form allows us to use the power rule for differentiation.

The power rule states that if we have a function $x^n$, its derivative with respect to $x$ is $n{x}^{n-1}$. Here, our exponent $n$ is $\frac{1}{2}$.

Applying the power rule to $x^{\frac{1}{2}}$, we get:

Next, we simplify the exponent $\frac{1}{2}-1$, which equals $-\frac{1}{2}$. Therefore, our expression for the derivative becomes:

To make this expression more familiar, we rewrite $x^{-\frac{1}{2}}$ using the square root notation. Since $x^{-\frac{1}{2}}$ is the same as $\frac{1}{x^{\frac{1}{2}}}$, and $x^{\frac{1}{2}}$ is $\sqrt{x}$, we get:

Substituting this back into our expression for the derivative, we have:

Therefore, the derivative of $\sqrt{x}$ with respect to $x$ is $\boxed{\frac{1}{2\sqrt{x}}}$.

Q.E.D.