Derivative of \( \displaystyle 5 \sqrt{\left(x - 1\right)^{2} + 1} \)

Problem 2.995 · medium

1. \[ \frac{d}{d x} 5 \sqrt{\left(x - 1\right)^{2} + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} \sqrt{\left(x - 1\right)^{2} + 1} \]
   constant-multiple rewritePull out the constant factor 5. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \left(\left(x - 1\right)^{2} + 1\right)}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{5 \frac{d}{d x} \left(x^{2} - 2 x + 2\right)}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]
   algebraExpand the inner quadratic expression.✓ Proved
5. \[ = \frac{5 \left(2 x - 2\right)}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]
   derivative algebraDifferentiate the inner polynomial. Factor out the 2 from the derivative.✓ Proved
6. \[ = \frac{5 x - 5}{\sqrt{\left(x - 1\right)^{2} + 1}} \]
   algebra simplifyMultiply the constants (5 * 1/2 * 2). Rewrite the negative exponent as a denominator.✓ Proved