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

Problem 2.1579 · hard

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