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

Problem 2.956 · medium

1. \[ \frac{d}{d x} \sqrt{\left(x + 1\right)^{2} + 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional power.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\left(x + 1\right)^{2} + 1\right)}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(x + 1\right)^{2}}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = \frac{\left(2 x + 2\right) \frac{d}{d x} \left(x + 1\right)}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]
   powerApply the power rule to (x + 1)**2.✓ Proved
5. \[ = \frac{2 x + 2}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]
   derivative simplifyDifferentiate the inner term (x + 1). Simplify the expression.✓ Proved
6. \[ = \frac{x + 1}{\sqrt{\left(x + 1\right)^{2} + 1}} \]
   simplify simplifyCancel the 2 and 1/2. Rewrite with a positive exponent.✓ Proved