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

Problem 2.1495 · hard

1. \[ \frac{d}{d x} \frac{5 \sqrt{4 x^{2} + 4 x + 2}}{2} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \sqrt{4 x^{2} + 4 x + 2}}{2} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \left(4 x^{2} + 4 x + 2\right)}{4 \sqrt{4 x^{2} + 4 x + 2}} \]
   chain algebraApply the chain rule. Distribute the power term.✓ Proved
4. \[ = \frac{5 \left(\frac{d}{d x} 2 + \frac{d}{d x} 4 x + \frac{d}{d x} 4 x^{2}\right)}{4 \sqrt{4 x^{2} + 4 x + 2}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{5 \left(8 x + 4\right)}{4 \sqrt{4 x^{2} + 4 x + 2}} \]
   derivative constantCompute the derivatives of the individual terms. The derivative of the constant 2 is 0.✓ Proved
6. \[ = \frac{10 x + 5}{\sqrt{4 x^{2} + 4 x + 2}} \]
   rewrite algebraRewrite the negative exponent as a denominator. Factor out 4 from (8x + 4) and simplify with 5/4.✓ Proved