Derivative of \( \displaystyle \frac{\sqrt{4 x^{2} + 1}}{2} \)

Problem 2.50 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{4 x^{2} + 1}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{4 x^{2} + 1}}{2} \]
   constant rewritePull out the constant factor 1/2. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(4 x^{2} + 1\right)}{4 \sqrt{4 x^{2} + 1}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} 4 x^{2}}{4 \sqrt{4 x^{2} + 1}} \]
   sumApply the sum rule to the inner function.✓ Proved
5. \[ = \frac{\frac{d}{d x} 4 x^{2}}{4 \sqrt{4 x^{2} + 1}} \]
   constantThe derivative of the constant 1 is 0.✓ Proved
6. \[ = \frac{2 x}{\sqrt{4 x^{2} + 1}} \]
   power algebra simplifyApply the power rule to 4*x**2. Move the negative exponent to the denominator. Simplify the fraction by dividing 8 by 4.✓ Proved