Derivative of \( \displaystyle \frac{\sqrt{16 x^{2} - 24 x + 10}}{2} \)

Problem 2.2145 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{16 x^{2} - 24 x + 10}}{2} \]
   constant-multipleStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{16 x^{2} - 24 x + 10}}{2} \]
   rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(16 x^{2} - 24 x + 10\right)}{4 \sqrt{16 x^{2} - 24 x + 10}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(16 x^{2} - 24 x\right)}{4 \sqrt{16 x^{2} - 24 x + 10}} \]
   constantDifferentiate the constant term 10.✓ Proved
5. \[ = \frac{- \frac{d}{d x} 24 x + \frac{d}{d x} 16 x^{2}}{4 \sqrt{16 x^{2} - 24 x + 10}} \]
   sumSplit the derivative into terms.✓ Proved
6. \[ = \frac{32 x - 24}{4 \sqrt{16 x^{2} - 24 x + 10}} \]
   power algebraDifferentiate the power terms. Combine the terms into a single fraction.✓ Proved
7. \[ = \frac{8 x - 6}{\sqrt{16 x^{2} - 24 x + 10}} \]
   simplifySimplify the fraction by dividing by 4.✓ Proved