Derivative of \( \displaystyle \frac{3 \sqrt{4 x^{2} + 8 x + 5}}{2} \)

Problem 2.1565 · hard

1. \[ \frac{d}{d x} \frac{3 \sqrt{4 x^{2} + 8 x + 5}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \sqrt{4 x^{2} + 8 x + 5}}{2} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(4 x^{2} + 8 x + 5\right)}{4 \sqrt{4 x^{2} + 8 x + 5}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = \frac{3 \left(\frac{d}{d x} 5 + \frac{d}{d x} 8 x + \frac{d}{d x} 4 x^{2}\right)}{4 \sqrt{4 x^{2} + 8 x + 5}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{3 \left(8 x + 8\right)}{4 \sqrt{4 x^{2} + 8 x + 5}} \]
   derivative simplifyCalculate the derivatives of each term. Simplify the derivative of the polynomial.✓ Proved
6. \[ = \frac{6 x + 6}{\sqrt{4 x^{2} + 8 x + 5}} \]
   algebra simplifyMove the negative exponent to the denominator. Distribute the constant and simplify the final expression.✓ Proved