Derivative of \( \displaystyle \frac{3 \sqrt{16 x^{2} - 8 x + 2}}{4} \)

Problem 2.1656 · hard

1. \[ \frac{d}{d x} \frac{3 \sqrt{16 x^{2} - 8 x + 2}}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \sqrt{16 x^{2} - 8 x + 2}}{4} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(16 x^{2} - 8 x + 2\right)}{8 \sqrt{16 x^{2} - 8 x + 2}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = \frac{3 \left(32 x - 8\right)}{8 \sqrt{16 x^{2} - 8 x + 2}} \]
   derivativeDifferentiate the inner polynomial.✓ Proved
5. \[ = \frac{3 \left(4 x - 1\right)}{\sqrt{16 x^{2} - 8 x + 2}} \]
   algebra algebraFactor out 8 from the polynomial. Cancel the 8 in the numerator and denominator.✓ Proved
6. \[ = \frac{12 x - 3}{\sqrt{16 x^{2} - 8 x + 2}} \]
   simplifyRewrite with a positive exponent in the denominator.✓ Proved