Question

solve the differential of root of x

Explanation:

🆕 New Concept Discovered: Power Rule for Differentiation
How to differentiate variables raised to any power.

Step 1: Rewrite the expression with a fractional exponent

To find the derivative of the square root of \(x\), we first rewrite the radical expression using a fractional exponent. The square root of \(x\) is the same as \(x\) raised to the power of \(\frac{1}{2}\).

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

Step 2: Apply the Power Rule

The Power Rule states that for any real number power \(n\):

$$\frac{d}{dx}(x^n) = n \cdot x^{n-1}$$

Applying this rule where \(n = \frac{1}{2}\):

$$\frac{dy}{dx} = \frac{1}{2} \cdot x^{\frac{1}{2} - 1}$$

Step 3: Simplify the exponent

Subtract \(1\) from the exponent:

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

This gives:

$$\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$$

Step 4: Convert back to radical form

A negative exponent means we take the reciprocal (move the term to the denominator), and the exponent \(\frac{1}{2}\) represents the square root:

$$\frac{dy}{dx} = \frac{1}{2 \cdot x^{\frac{1}{2}}} = \frac{1}{2\sqrt{x}}$$

Answer:

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