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

Problem 2.1658 · hard

1. \[ \frac{d}{d x} \sqrt{4 x^{2} + 8 x + 5} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional exponent.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(4 x^{2} + 8 x + 5\right)}{2 \sqrt{4 x^{2} + 8 x + 5}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 5 + \frac{d}{d x} 8 x + \frac{d}{d x} 4 x^{2}}{2 \sqrt{4 x^{2} + 8 x + 5}} \]
   sumDifferentiate the sum term by term.✓ Proved
4. \[ = \frac{8 x + 8}{2 \sqrt{4 x^{2} + 8 x + 5}} \]
   derivative simplifyDifferentiate each term inside the parentheses. Simplify the expression.✓ Proved
5. \[ = \frac{4 x + 4}{\sqrt{4 x^{2} + 8 x + 5}} \]
   algebra simplifyDistribute the 1/2 into the parentheses. Rewrite the negative exponent as a fraction.✓ Proved