Derivative of \( \displaystyle \frac{\sqrt{4 x^{2} - 12 x + 10}}{2} \)

Problem 2.1608 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{4 x^{2} - 12 x + 10}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{4 x^{2} - 12 x + 10}}{2} \]
   constant-multiple rewritePull out the constant factor 1/2. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(4 x^{2} - 12 x + 10\right)}{4 \sqrt{4 x^{2} - 12 x + 10}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{\frac{d}{d x} 10 - \frac{d}{d x} 12 x + \frac{d}{d x} 4 x^{2}}{4 \sqrt{4 x^{2} - 12 x + 10}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{8 x - 12}{4 \sqrt{4 x^{2} - 12 x + 10}} \]
   derivative algebra algebraDifferentiate each term in the polynomial. Simplify the derivative of the polynomial. Combine the terms into a single fraction.✓ Proved
6. \[ = \frac{2 x - 3}{\sqrt{4 x^{2} - 12 x + 10}} \]
   simplifySimplify the fraction by dividing the numerator and denominator by 4.✓ Proved