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

Problem 2.1738 · hard

1. \[ \frac{d}{d x} 2 \sqrt{x^{2} + 2 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 2 \frac{d}{d x} \sqrt{x^{2} + 2 x + 2} \]
   constant-multiple rewritePull out the constant factor 2. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(x^{2} + 2 x + 2\right)}{\sqrt{x^{2} + 2 x + 2}} \]
   chain algebraApply the chain rule. Simplify the constant factors.✓ Proved
4. \[ = \frac{2 x + 2}{\sqrt{x^{2} + 2 x + 2}} \]
   derivative algebra simplifyDifferentiate the inner polynomial. Rewrite with a positive exponent. Factor out the 2 in the numerator.✓ Proved