Question

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

• A. \( \frac{2}{\sqrt{x}} \)
• B. \( \frac{1}{2\sqrt{x}} \)
• C. \( \frac{1}{\sqrt{x}} \)
• D. \( -\frac{1}{\sqrt{x}} \)

Hint:

For powers of \( x \), use the power rule \( \frac{d}{dx} x^n = n x^{n-1} \), and adjust the exponent accordingly.

Answer 1

The Correct Option is C

We are asked to differentiate \( 2\sqrt{x} \), or \( 2x^{\frac{1}{2}} \). Step 1: Apply the power rule
The power rule for differentiation is given by: \[ \frac{d}{dx} x^n = n x^{n-1} \] We can treat \( 2\sqrt{x} \) as \( 2x^{\frac{1}{2}} \). Differentiating using the power rule: \[ \frac{d}{dx} \left( 2x^{\frac{1}{2}} \right) = 2 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = x^{-\frac{1}{2}} \] This simplifies to: \[ \frac{1}{\sqrt{x}} \] Thus, the derivative of \( 2\sqrt{x} \) is \( \frac{1}{\sqrt{x}} \).