Derivative of \( \displaystyle 3 \sqrt{\left(x + 2\right)^{2} + 1} \)

Problem 2.155 · medium

1. \[ \frac{d}{d x} 3 \sqrt{\left(x + 2\right)^{2} + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 3 \frac{d}{d x} \sqrt{\left(x + 2\right)^{2} + 1} \]
   constant-multiple rewritePull the constant factor out of the derivative. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)}{2 \sqrt{\left(x + 2\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{3 \left(2 x + 4\right)}{2 \sqrt{\left(x + 2\right)^{2} + 1}} \]
   derivative simplifyDifferentiate the inner function. Simplify the derivative of the inner term.✓ Proved
5. \[ = \frac{3 x + 6}{\sqrt{\left(x + 2\right)^{2} + 1}} \]
   algebra rewriteCancel the 2 and the 1/2. Rewrite the negative exponent as a fraction.✓ Proved